\begin{Definition}\label{def:phi homogenization}
Fix $\lambda = \lambda(n) > 0$ sufficiently small such that $\sigma = C_1(n)\lambda$ satisfies $4^{n+1}\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \ref{prop:basic}. For $\Lambda\ge \max\{\Lambda_0,\Lambda_1,\Lambda_2\}$, and any nonnegative functions $v \in C(\Omega)$, we define its $\phi$-homogenization $M_{\phi,\Lambda}v(x):=M_{\phi,\lambda,\Lambda}v(x)$ in $\frac{7}{8}\Omega$ through the following maximal averaging type operator:
\begin{equation}\label{eq:phi homogenization}
\begin{split}
M_{\phi,\Lambda}v(x)=  \sup\left\{ t\in\R:\; \exists \ \check{h}(x)<h<  \hat{h}(x),\ s.t., \
\frac{\mu ( \{v> t\} \cap S_h (x)  )}{\mu(S_h (x))} > \lambda \right\}.
\end{split}
\end{equation}
\end{Definition}

\begin{Theorem}\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying \eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \eqref{eq:periodic equation ma} with quadratic component $P$.
\end{Theorem}

In this paper, we study convex solutions to the Monge-Amp\`ere equation
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \mathbb{R}^n,
\end{equation}
where $\mu\not\equiv 0$ is a nonnegative locally finite Borel measure on $\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\mu $ is periodic with respect to the integer lattice $\mathbb{Z}^n := \{(k_1,\dots,k_n) \mid k_i \in \mathbb{Z}\} \subset \mathbb{R}^n$, i.e., for any Borel set $E \subset \mathbb{R}^n$,
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$.
\end{Definition}

Note that if $u$ is a periodic solution to \eqref{eq:periodic equation ma} and is expressed as \eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
where $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$.

Our first result is the uniqueness of periodic solutions.

\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}

Let $\mu \not\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\R^n$. We show that any convex solution to the Monge-Amp\`ere equation
\[
\det D^2 u = \mu \quad \text{in } \R^n
\]
admits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,
in full generality, the earlier works for the case $\mu=f(x)\,\ud x$: when $\log f \in C^\alpha$, it was established by Caffarelli and Li; and
when $\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures.

Let $\mathcal{S}_+^{n\times n}$ denote the set of positive definite symmetric $n\times n$ matrices.
\begin{Definition}[Periodic solution]\label{def:periodic solution}
A \emph{periodic solution} to \eqref{eq:periodic equation ma} is a convex function $u \in C(\mathbb{R}^n)$ satisfying \eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as
\begin{equation}\label{eq:periodic solution}
u(x) = v(x) +P(x),
\end{equation}
where $v$ is periodic with respect to $\mathbb{Z}^n$ and $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ is a quadratic function with $A\in \mathcal{S}_+^{n\times n}$, $b\in\R^n$ and $c\in\R$.
\end{Definition}

Our first result is the uniqueness of periodic solutions.

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof. For the uniqueness, when $\mu=f(x)\,\ud x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

\begin{Proposition}\label{prop:semiconcave}
Let $u$ be a convex solution to \eqref{eq:periodic equation ma} with $\mu$ being a $\mathbb{Z}^n/8n$-periodic measure (where $\mathbb{Z}^n := \Gamma_{\{e_1,\cdots,e_n\}}$) satisfying $\mu(\mathbb{Z}^n) = 1$.
Under the nondegeneracy condition
\begin{equation}\label{eq:nondegenerate}
\Delta^2_{i} u(0):=\Delta^2_{e_i} u(0) > 0 \quad \text{for all } 1 \leq i \leq n,
\end{equation}
there exists a unique compatible matrix $A$ satisfying:
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az ,\quad \forall z \in \mathbb{Z}^n.
\]
\end{Proposition}
\begin{proof}
Without loss of generality, we may assume $u(0) = 0$ and $u \geq 0$.
Let us define $S_h := \{x \in \mathbb{R}^n : u(x) \leq h\}$, and select $\Lambda_u>0$ such that the section $S_{\Lambda_u}$ satisfies $\kappa_{\mathbb{Z}^n}(S_{\Lambda_u}) \geq \Lambda_0^2$.
Let us now normalize $S_h$ such that $B_1(0) \subset T_h^{-1} S_h \subset B_n(0)$ and consider the normalized function
\[
u_{h}(x) =\frac{u(T_h x)}{(\det T_h)^{\frac{2}{n}}} \quad \text{with } \det D^2 u_h= \mu_h=\mu \circ T_h .
\]
Due to the periodicity of $\mu$ and large $h$, we know that
\[
c(n)h^{\frac{n}{2}} \leq \det T_h \leq C(n)h^{\frac{n}{2}},
\]
and thus, $c(n)h^{\frac{n}{2}} \leq |S_h| \leq C(n)h^{\frac{n}{2}}$. Applying Proposition \ref{prop:basic} to $\phi=u_h-h(\det T_h)^{-\frac{2}{n}}$, we obtain that for sufficiently large $h$,
\[
c(n) (h/\Lambda_u)^{\frac{1}{3}} S_{\Lambda_u} \subset S_h \subset C(n) (h/\Lambda_u)^{\frac{2}{3}} S_{\Lambda_u},
\]
and consequently, $B_{c(n) h^{\frac{1}{4}}}\subset S_h\subset B_{C(n) h^{\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have
\[
\quad c(n)h^{\frac{1}{4}} I\leq T_h \leq C(n)h^{\frac{3}{4}} I.
\]
The transformation law \eqref{eq:transformation law} then yields the following estimates at corresponding points
\begin{equation}\label{eq:transformh}
c(n)h^{-\frac{1}{2}}\Delta^2_{e_i} u \leq \Delta^2_{T_h^{-1} e_i} u_h \leq C(n)h^{\frac{1}{2}}\Delta^2_{e_i} u.
\end{equation}

\textbf{Step 4.} Suppose for some sequence $h \to \infty$, the rescaled functions $\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \frac{1}{2}x^{\top}A x$ for $A \in \mathcal{S}_+^{n\times n}$ with $\det A = 1$.
For each fixed direction $i$, we denote
\[
\alpha=\sup_{\R^n}\Delta^2_{e_i} u,\quad \beta= e_i^{\top} A e_i.
\]
Note that the estimate \eqref{eq:w21 estimate} continues to hold for $\tilde{u}_{h}$, from which we derive $\alpha \geq \beta$.
We claim
\begin{equation}\label{eq:w2infty sup}
\alpha=\beta.
\end{equation}
Assume by contradiction that $\alpha = \beta + 4s$ for some $s > 0$.
Through rescaling from infinity, we may assume without loss of generality that there exist points $x_h \in B_{1/8}$ satisfying
\[
\alpha- \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h}(x_h)= \inf_{B_{1/8}} (\alpha- \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_{h} ) \leq a_h,
\]
where $a_h \to 0 $ as $h \to \infty$. Define the normalized function $v = (\alpha - \Delta^2_{h^{\frac{1}{2}}e_i} \tilde{u}_{h})/a_h$ and set $\delta = s/[C(\beta + 2s)]$. Applying Lemma \ref{lem:Theorem 7.3.1 modify} yields with $\phi=\tilde{u}_h-1$ and $\Omega=S_{1,h}=\{\tilde{u}_h \leq 1\} $, we obtain that
\[
\tilde{\mu}_h ( \{v\leq M_1(n,\delta) \} \cap S_{1,h} ) \geq (1-\delta) \tilde{\mu}_h(S_{1,h} ),
\]
where $\tilde{\mu}_h=\tilde{f}_{h}\,\ud x$. For sufficiently small $a_h$, we consequently obtain
\[
\tilde{\mu}_h\left(\left\{\Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h>\beta +2 s\right\} \cap S_{1,h}\right)= \tilde{\mu}_h\left(\left\{v <\frac{2s}{a_h}\right\} \cap S_{1,h}\right)\geq (1-\delta) \tilde{\mu}_h (S_{1,h}).
\]
This leads to the integral lower bound
\[
\int_{S_{1,h}} \Delta^2_{h^{\frac{1}{2}} e_i} \tilde{u}_h \ud \tilde{\mu}_h \geq (1-\delta) (\beta+2s) \tilde{\mu}_h (S_{1,h})\geq
(\beta+s) \tilde{\mu}_h (S_{1,h}).
\]
This contradicts the uniform estimate \eqref{eq:w21 estimate} after performing a suitable rescaling.

\begin{proof}[Proof of Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}]
Let $u$ be a global solution of \eqref{eq:periodic equation ma}.
Clearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\mu$ is periodic and $\mu \not\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \subset B_R(0)$.
We now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \eqref{eq:nondegenerate} in Proposition \ref{prop:semiconcave} is satisfied. Hence, by Proposition \ref{prop:semiconcave}, there exists a compatible matrix $A \in \mathcal{S}_+^{n \times n}$ such that
\[
\sup_{ \R^n}\Delta_z^2 u=z^{\top}Az,\quad \forall z \in \mathbb{Z}^n.
\]
Let $w$ be a periodic solution to \eqref{eq:periodic equation ma} in the sense of Definition \ref{def:periodic solution} with quadratic part $\frac{1}{2}z^{\top} A z $. Then $\Delta_z^2 w=z^{\top}Az$. Let
\[
v(x)=u(x)-w(x).
\]
By subtracting a suitable linear function, we now assume that $v(0)=0$ and
\[
v(e_i)=v(-e_i),\quad 1\leq i\leq n.
\]
Noting that $\sup_{\R^n} \Delta_z^2 v = \sup_{\R^n} \Delta_z^2 (u - w) = 0$, we then find that
\[
v(\pm ke_i)\leq 0 \quad \text{for all } k\in \mathbb{Z},\ 1\leq i\leq n.
\]

The existence of periodic solutions follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof.  For the uniqueness, when $\mu=f(x)\,\mathrm{d} x$ is a positive measure, the case $\log f \in C^\alpha$ was proved by Li \cite{li1990existence}, and the case  $\log f\in L^\infty$ was shown by Li-Lu \cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.

Our second result is the classification below, which extends the main results  of Caffarelli-Li \cite{caffarelli2004liouville} and Li-Lu \cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \cite{li2019monge}.

\begin{Theorem}\label{thm:solutions are periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying \eqref{eq:periodic f}, and let $u$ be a convex solution of \eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \ref{def:periodic solution}.
\end{Theorem}

Caffarelli-Li \cite{caffarelli2004liouville} first proved Theorem \ref{thm:solutions are periodic} for positive measures $\mu=f(x)\,\mathrm{d} x$ with $\log f \in C^\alpha$, and conjectured the extension to merely bounded $\log f$. This conjecture was resolved by Li-Lu \cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \geq 0$ with $f \not\equiv 0$. Theorem \ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.

\begin{Remark}
When $\mu \equiv 0$, equation \eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\`ere equation $\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \cite{caffarelli1986dirichlet}).
In particular, writing $x = (x',x_n) \in \mathbb{R}^{n-1} \times \mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.
\end{Remark}

Since $\mu$ is assumed to be nonnegative,  equation \eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\`ere equation, for which many arguments in  \cite{caffarelli2004liouville,li2019monge} do not apply.  The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or  Besicovitch covering lemma which is essential for the Calder\'on-Zygmund decomposition on sections.
Therefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez \cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\`ere equations.

Nonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\infty}$ bounds fail, Theorem \ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially. Finally, we obtain  in Theorem \ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.

After overcoming this critical difficulty arising from the degeneracy of equation \eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \cite{caffarelli2004liouville,li2019monge} for positive measures $\mu=f(x)\mathrm{d} x$ to  the degenerate case considered here.  In this step, we must also overcome additional difficulties due to the degeneracy of
 \eqref{eq:periodic equation ma}. Consequently, when Theorem \ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow.

This paper is organized as follows. In Section \ref{sec:harnack}, we study convex functions whose Monge-Amp\`ere measure has small period, establishing a dichotomous Harnack type  inequality for linearized Monge-Amp\`ere equations. Section \ref{sec:semiconvavity} investigates  the uniqueness of compatible quadratic component of each solution in the sense of Proposition \ref{prop:semiconcave}. Finally, in Section \ref{sec:main theorem}, we prove  Theorems \ref{thm:uniqueness within periodic} and \ref{thm:solutions are periodic}.

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while Theorem~\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\ref{thm:harnack super}); for subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\ref{thm:harnack sub}): “either the $L^{\infty}$ bound holds or the $L^{\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\ref{thm:harnack}).

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”

The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\'errez.”

To overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:
\begin{Theorem}\label{thm:harnack super}
Assume that $v \in C^2(\Omega)$ is a nonnegative supersolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then there exist positive constants  $\Lambda_6(n)$, $\epsilon(n) $ and $C(n) $ such that if
\[
\inf_{B_{1/4}} v \leq 1,
\]
then for $\Lambda\ge \Lambda_6(n)$, we have
\begin{equation}\label{eq:harnack super}
\mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})< Ct^{-\epsilon} .
\end{equation}
\end{Theorem}
For subsolutions (where “standard $L^{\infty}$ bounds fail”), they prove the following “dichotomy”:
\begin{Theorem}\label{thm:harnack sub}
Assume that $v \in C^2(\Omega)$ with $v \geq 0$ is a subsolution of \eqref{eq:linearized ma phi} in $\Omega$.
Then, there exist positive constants $\Lambda_7(n)$ and $\beta(n)$ such that for any $\epsilon > 0$, there is a constant $C_2(n, \epsilon)$ with the following property: If $\Lambda\ge \Lambda_7(n)$ and
\[
\min\{\mu(\{v >  t\}  \cap B_{1/2}), \mu(\{M_{\phi,\Lambda}v >  t\} \cap B_{1/2})  \} \leq t^{-\epsilon}   \quad  \forall t>0,
\]
then either
\[
\sup_{B_{1/4} } v  \leq C_2(n,\epsilon),
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2} } v  \geq e^{\beta \kappa^{\frac{1}{3}} } \sup_{B_{1/4} }v.
\]
\end{Theorem}
Combining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:
\begin{Theorem}\label{thm:harnack}
Let $\Omega$ and $\widetilde{\Omega}$ be open convex subsets of $\mathbb{R}^n$ satisfying $B_1\subset \Omega \subset B_n, B_1\subset \widetilde{\Omega} \subset B_n$, $n\geq 2$, and let $\phi\in C^2(\overline{\Omega})$ and $\tilde{\phi}\in C^2(\overline{\widetilde{\Omega}})$ be convex functions satisfying,
\[
\mu_\phi =\mu \quad \text{in } \Omega, \quad \phi=0 \quad \text{on } \partial \Omega,
\]
\[
\mu_{\tilde{\phi}} =\mu \quad \text{in }  \widetilde{\Omega}, \quad \tilde{\phi}=0 \quad \text{on } \partial \widetilde{\Omega},
\]
where $\mu$ is a periodic Borel measure defined on $\R^n$ satisfying \eqref{eq:period gamma},  \eqref{eq:normalized assumption}, and \eqref{eq:small period assumption}. Assume that $v\in C^2(\Omega) \cap C^2(\widetilde{\Omega})$ with $v\geq 0$ satisfies
\[
L_{\phi}v \leq 0   \quad  \text{in } \Omega, \quad L_{\tilde{\phi}}v \geq 0   \quad  \text{in }\widetilde{\Omega}.
\]

Then there exist positive constants $\beta(n)$, $\Lambda_9(n)$,  and $C(n)$ such that if $\Lambda>\Lambda_9(n)$, then either
\[
\sup_{B_{1/4}}v\leq C\inf_{B_{1/4}}v,
\]
or for $\kappa =\kappa(B_1)\geq \Lambda^2$ that
\[
\sup_{B_{1/2}} v  \geq e^{\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/4}}v.
\]
Consequently, we have
\[
\sup_{B_{1/4}}v \leq C\inf_{B_{1/4}}v+e^{-\beta \kappa^{\frac{1}{6}}}\sup_{B_{1/2}} v.
\]
\end{Theorem}

“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.

\label{thm:uniqueness within periodic}
Let $\mu \not\equiv 0$ be a nonnegative locally finite Borel measure satisfying
\begin{equation}\label{eq:periodic f}
\mu (E+z)=\mu (E),\quad \forall \ z \in \mathbb{Z}^n.
\end{equation}
and let $P(x)=\frac{1}{2}x^{\top}Ax+b\cdot x+c$ be a quadratic function with $A\in \mathcal{S}_+^{n\times n}$ satisfying
\begin{equation}\label{eq:A compatible}
 \det A =\mu (\mathbb{T}^n),
\end{equation}
Then, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to
\begin{equation}\label{eq:periodic equation ma}
\det D^2 u = \mu \quad \text{in } \R^n,
\end{equation}
with quadratic component $P$.

Question: Let $\mu\not\equiv 0$ be a nonnegative locally finite Borel measure on $\mathbb{R}^n$ that is $\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\subset \mathbb{R}^n$ and every $z\in \mathbb{Z}^n$,\n\[\n\mu(E+z)=\mu(E).\n\]\nLet\n\[\nP(x)=\tfrac12 x^\top A x+b\cdot x+c,\n\]\nwhere $A\in \mathcal S_+^{n\times n}$ is positive definite symmetric, $b\in \mathbb{R}^n$, $c\in \mathbb{R}$, and\n\[\n\det A=\mu(\mathbb{T}^n),\qquad \mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n.\n\]\nA periodic solution of\n\[\n\det D^2u=\mu\quad\text{in }\mathbb{R}^n\n\]\nmeans a convex function $u\in C(\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?

There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.

Choices:
B. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\mathbb{R}^n$.
C. There exists at most one periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\mathbb{R}^n$.
D. For every quadratic function $P(x)=\tfrac12 x^\top A x+b\cdot x+c$ with $A\in \mathcal S_+^{n\times n}$ satisfying $\det A\ge \mu(\mathbb{T}^n)$, there exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants.
E. There exists a periodic solution $u$ of $\det D^2u=\mu$ in $\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\mathbb{R}^n$.

constant-shift ambiguity in comparing two solutions

dropped existence while retaining uniqueness up to constants

exact compatibility condition $\det A=\mu(\mathbb{T}^n)$

normalization removing linear parts before applying Harnack to the difference


\begin{theorem}\label{thm:main1}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.
\end{theorem}

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} $T(X) \subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

\begin{abstract}
Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.
\end{abstract}

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

\begin{theorem}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ to a free abelian group of countably infinite rank.
\end{theorem}
\begin{proof}
Let $\mathbb{Z}[\mathcal{S}(X_n)] = \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n)} \mathbb{Z}$. Elements of $\mathbb{Z}[\mathcal{S}(X_n)]$ can be regarded as functions $\mathcal{S}(X_n) \to \mathbb{Z}$ with finite support. Define a homomorphism $\Phi' : T(X_n) \to \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n) } \mathbb{Z}$ by taking $(\Phi'(f))(\mathfrak{s}) = SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$. The function $\mathfrak{s} \mapsto SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.

The image $im(\Phi')$ of $\Phi'$ is a subgroup of the free abelian group $\mathbb{Z}[\mathcal{S}(X_n)]$ and hence is free abelian. Let $\Phi : T(X_n) \to im(\Phi')$ be the homomorphism obtained by factoring $\Phi'$ through its image. The theorem will follow if we can show that $im(\Phi')$ has infinite rank (since $\mathbb{Z}[\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\Phi')$ can be at most countably infinite). For $n=10$, Lemma \ref{lem:torelli} immediately implies that $im(\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \# \overline{\mathbb{CP}^2}$. Let $f \in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \# id_{\overline{\mathbb{CP}^2}} \in T(X_n)$. Let $\kappa$ be a spin$^c$-structure on $\overline{\mathbb{CP}^2}$ with $c(\kappa)^2 = -1$. Let $\mathfrak{s} \in \mathcal{S}(X_{n-1})$. Then $\mathfrak{s} \# \kappa \in \mathcal{S}(X_n)$. The blowup formula gives
\[
SW^c_{X_n , \mathfrak{s} \# \kappa , \mathbb{Z}}(f \# id_{\overline{\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \mathfrak{s} , \mathbb{Z}}(f).
\]
Hence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \mathfrak{s} , \mathbb{Z}} : T(X_n) \to \mathbb{Z}$ is non-zero and hence $\Phi'( T(X_n) )$ has infinite rank.
\end{proof}

\begin{theorem}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}
\begin{proof}
Define $\Phi' : M_+(X) \to \mathbb{Z}^\infty$ to be given by
\[
\Phi' = \bigoplus_{n=1}^{\infty} SW_{X , \mathcal{O}_{2n-1}} : M_+(X) \to \bigoplus_{n=1}^{\infty} \mathbb{Z}.
\]
To prove the result, it suffices to show that the image of $\Phi'$ is not finitely generated. Then we let $\Phi$ be the homomorphism obtained by replacing the codomain of $\Phi'$ with the image of $\Phi'$. The argument is almost identical to the proof of \cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.
\end{proof}

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

Our second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

Note that by Schreier's lemma \cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.

The first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \cite{bar} and Konno \cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ where $n \ge 3$ is odd and also for $X = n (S^2 \times S^2) \# nK3$, $n \ge 1$. Note that these are precisely the manifolds of the form $E(m) \# (S^2 \times S^2)$, $m \ge 2$. Theorem \ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \# (S^2 \times S^2) = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$.

\subsection{Outline of the proofs of the main results}
To each diffeomorphism $f \in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \times X$ by identifying the ends via $f$. If $\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:
\begin{itemize}
\item[(1)]{{\bf The constant chamber:} assume that $f \in T(X)$. Then $f$ acts trivially on $H^2(X ; \mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\mathcal{H}^+ \to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\mathcal{H}^+$ under the above trivialisation.}
\item[(2)]{{\bf The zero chamber:} assume that $c(\mathfrak{s})^2 \ge 0$ and $c(\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}
\end{itemize}

These two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \mathfrak{s}}(f)$ and $SW^0_{X , \mathfrak{s}}(f)$ depending only on $(X , \mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \mathfrak{s}} : T(X) \to \mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \in T(X)$, the invariants $SW^c_{X , \mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism
\[
\Phi : T(X) \to \bigoplus_{\mathfrak{s}} \mathbb{Z}
\]
where the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \ref{thm:main1} follows by showing that the image of $\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\{ \mathfrak{s}_n\}$ and diffeomorphisms $\{ t_n \}$ for which $SW^c_{X , \mathfrak{s}_n}( t_n ) \neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \cite{rub1}, \cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \# (S^2 \times S^2) \cong E(1) \# (S^2 \times S^2)$.

\noindent{\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.

To prove Theorem~\ref{thm:main1}, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of Theorem~\ref{thm:main1} then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.

To prove the main theorem, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.,

Question: For each integer $n \ge 10$, let $X_n = 2\mathbb{CP}^2 \# n\overline{\mathbb{CP}^2}$. Let the mapping class group $M(X_n)$ be the group of smooth isotopy classes of diffeomorphisms of $X_n$, and let the Torelli group $T(X_n) \subseteq M(X_n)$ be the subgroup consisting of smooth isotopy classes of diffeomorphisms that are continuously isotopic to the identity. Here $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank. Which existence statement holds for these manifolds?

For every $n \ge 10$, there exists a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.

Choices:
B. For every $n \ge 10$, there exists a surjective homomorphism $\Phi : M(X_n) \to \mathbb{Z}^\infty$.
C. For every $n \ge 10$, there exists a homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ with infinite image.
D. For every $n \ge 10$, there exists a single homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ that is defined uniformly for all $n$ and is surjective for each $X_n$.
E. For every $n \ge 9$, there exists a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.

constant-chamber homomorphisms require Torelli trivial action on cohomology

dropped surjectivity onto all of \mathbb{Z}^\infty, keeping only infinite image

dependence of the construction on the individual manifold X_n


\begin{proposition}\label{pr:q.discont.act.join}
  The free diagonal right action of $\left(\bQ,+\right)$ on $\bQ*\bQ$ is discontinuous if the join is equipped with the quotient topology.
\end{proposition}

\begin{theorem}\label{th:lb.cc}
  For Hausdorff topological groups $\bG$ with $\bG^n$ countably-compact the $\bG$-action on the universal space $\left(E_n\bG,\tau_{\varinjlim}\right)$ is continuous.
\end{theorem}

\begin{theorem}\label{th:disc.cone}
  If a Hausdorff topological group $\bG$
  \begin{itemize}[wide]
  \item acts discontinuously on its cone $\cC \bG$ with the quotient topology;

  \item or has distinct quotient and product topologies on $\bG\times \cC \bG$
  \end{itemize}
  then it acts discontinuously on its quotient-topologized join $\bG*\bG$, and hence also on $E_n \bG$, $1\le n\le \infty$.  \qedhere
\end{theorem}

\label{se:colim.top}

Specializing \Cref{pr:princ.bdl.ext} back to trivial $\bU$, the total space $E\bG=E_{\alpha}\bG$ featuring in that proof for a (here, compact) topological group $\bG$ is variousl

\begin{equation}\label{eq:x.ast.y}
  X*Y
  :=
  X\times Y\times [0,1]
  \bigg/
  \left(
    \begin{aligned}
      (x,y,0)&\sim (x,y',0)\\
      (x,y,1)&\sim (x',y,1)\\
    \end{aligned}
  \right)
\end{equation}

\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:
\begin{itemize}[wide]
\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \emph{locally trivial} \cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:
  \begin{itemize}[wide]
  \item We freely (and sometimes tacitly) pass back and forth \cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\times q$-matrix algebra bundles with their corresponding \emph{principal bundles} \cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\left(\text{central }\bC^{\times}\right)$ respectively.

  \item It is harmless by the selfsame \cite[Assertion 18.3.4]{hjjm_bdle} to furthermore work with \emph{Hermitian} vector bundles, i.e. \cite[\S 14.1]{ms-cc} those equipped with appropriately compatible Hilbert-space structures on their fibers. In principal-bundle language this has the effect of reducing the groups to $U(q)<GL(q)$ and $PU(q)<PGL(q)$ respectively.
  \end{itemize}

\item On the other hand, there are the \emph{Banach bundles}
  \begin{equation*}
    \cA
    \xrightarrowdbl[\quad\text{continuous open}\quad]{\quad\pi\quad}
    X
    ,\quad
    \text{Banach-space fibers }\left(\cA_x:=\pi^{-1}x,\ \|\cdot\|_x\right)
  \end{equation*}
  of \cite[Definition II.13.4]{fd_bdl-1}, \cite[Definition 1.1]{dg_ban-bdl}, \cite[\S 3]{zbMATH03539905} and recalled below in \Cref{def:eqv.ban.bdl}. They \emph{can} be locally trivial in the sense of \cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \emph{continuous} in the sense that the globally-defined norm
  \begin{equation}\label{eq:glob.nrm}
    \cA
    \xrightarrow{\quad\|\cdot\|\quad}
    \bR_{\ge 0}
    ,\quad
    \|\cdot\||_{\cA_x}
    :=
    \|\cdot\|_x
  \end{equation}
  is continuous (\emph{(F)} Banach bundles in \cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).

  In full generality (the \emph{(H)} Banach bundles of loc. cit.) \Cref{eq:glob.nrm} would be only \emph{upper semicontinuous} \cite[\S 1.3]{hk_shv-bdl}. Bundle base spaces and topological groups are always assumed Hausdorff absent explicit warnings.
\end{itemize}

Per standard conventions (e.g. \cite[Definition 2.2]{dupre_hilbund-1} or \cite[Definition IV.1.4.1]{blk} in a slightly different but related setting), Banach bundles are \emph{(sub-)homogeneous} if all fibers have the same finite dimension (or, respectively, there is an upper finite bound on fiber dimensions). For occasional added specificity, \emph{$n$-homogeneity} and \emph{$n$-sub-homogeneity} refer to the fibers being $n$ or $(\le n)$-dimensional respectively.

By way of a recollection before stating some of the main results, we reprise the usual Banach-bundle definitions in a slightly expanded framework to afford \emph{equivariance} with respect to a group operating on the entire package, as this will be the framework of interest throughout the paper (cf. \cite[Definition 1.1]{dg_ban-bdl}).

\begin{definition}\label{def:eqv.ban.bdl}
  For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
  \begin{itemize}[wide]
  \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

  \item with the \emph{fibers} $\cE_x:=\pi^{-1}x$, $x\in X$ carrying Banach-space structures compatible with the $\bU$-actions (which thus restrict to jointly-continuous representations $\bU\times \cE_x\to \cE_x$);

  \item so that scalar multiplication and addition are continuous;

  \item and the norm on $\cE$ is continuous (for \emph{continuous} or \emph{(F)} Banach bundles) or upper semicontinuous (for \emph{(H)} Banach bundles).
  \end{itemize}
  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\cE_x$ carrying those richer structures instead.
\end{definition}

The common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.
\begin{enumerate}[(a),wide]
\item\label{item:motiv.brnch} The \emph{non-commutative branched covers} introduced in \cite[Deﬁnition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\le A$ which satisfying convenient finiteness conditions (being of \emph{finite index} \cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \cite{bg_cx-exp,2409.17807v1,2409.03531v2}.

  It is in that setting that the embeddability of a subhomogeneous $C^*$ bundle into a homogeneous one becomes relevant: homogeneity (hence also homogeneous embeddability) will ensure \cite[Proposition 3.4]{bg_cx-exp} the desired finiteness constraints hold, whereas they need not if the subhomogeneous bundle's fiber-dimension jump loci are sufficiently ill-behaved (per \cite[Theorem 1.3]{2409.17807v1} addressing \cite[Problem 3.11]{bg_cx-exp}, say).

\item\label{item:motiv.sph} An adjacent theme to embeddability into homogeneous bundles is their \emph{prolongation} from closed subsets of the base space; this is what the title's \emph{germs} refer to: the term usually refers to objects (sheaf sections \cite[p.2]{bred_shf_2e_1997}, functions \cite[\S 27.18]{km_glob}) defined locally around a target (point, closed subset, etc.), identified if in agreement on a sufficiently small neighborhood of that target.

  One instance of the problem (which did in fact provide motivation for the present work) arises in relation to studying the \emph{non-commutative spheres} $\bS_{\theta}^{2n-1}$ introduced in \cite[Definition 2.1]{no_sph} as objects dual to the unital $C^*$-algebras generated universally by
  \begin{equation*}
    \text{normal }T_k,\quad k\in \left\{1..n\right\}
    \quad:\quad
    \left[
    \begin{aligned}
      T_{\ell} T_k
      &=
        \exp\left(2\pi i \theta_{k\ell}\right) T_k T_{\ell},\quad\forall k,\ell\\
      \sum_{k=1}^n T_k^* T_k
      &=1
    \end{aligned}
    \right.
  \end{equation*}
  for skew-symmetric $\theta\in M_n(\bR)$. The latter is the same type of deformation parameter featuring in the definition of the \emph{non-commutative tori} $\bT^n_{\theta}$ (\cite[p.193]{rief_case}, \cite[\S 12.2]{gbvf_ncg}), the difference being that for these the generators $U_k$ would be unitary rather than normal.

  For rational deformation parameters $\theta\in M_n(\bQ)$ the $\bS^{2n-1}_{\theta}$ are non-commutative branched covers over classical spheres $\bS^{2n-1}$ in the sense of \Cref{item:motiv.brnch} (e.g. by a conjunction of \cite[Theorem B]{2508.04922v1} and \cite[Theorem A]{2409.03531v2}), particularly pleasant for $n=2$ \cite[Proposition 4.5]{cp_cont-equiv_xv3}:
  \begin{equation*}
    \cA\xrightarrowdbl{\quad \pi\quad}\bS^3
    ,\quad
    \cA_x\cong
    \begin{cases}
      M_q&x\not\in h^{-1}\left(p_{\pm}\right)\\
      \bC^q&\text{otherwise},
    \end{cases}
    ,\quad
    \begin{aligned}
      \bS^3
      &\xrightarrowdbl[\ \text{Hopf fibration}\ ]{\quad h\quad}\bS^2\\
      p_{\pm}&:=\text{antipodal pair}
    \end{aligned}
  \end{equation*}
  where $q$ is the lowest-terms denominator of the rational $\theta_{12}$.

  It is useful, in computing the dimension invariants attached by \cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \cite[Deﬁnition 2.3]{cpt_dim}) to finite-group actions on $\bS^{2n-1}_{\theta}$, to extend the $\bC^q$-fibered restriction of $\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \Cref{th:prlng.homog.abdls} affords this.
\end{enumerate}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:
\begin{itemize}[wide]
\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \emph{locally trivial} \cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:
 \begin{itemize}[wide]
 \item We freely (and sometimes tacitly) pass back and forth \cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\times q$-matrix algebra bundles with their corresponding \emph{principal bundles} \cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\left(\text{central }\bC^{\times}\right)$ respectively.

\item On the other hand, there are the \emph{Banach bundles}
 \begin{equation*}
 \cA
 \xrightarrowdbl[\quad\text{continuous open}\quad]{\quad\pi\quad}
 X
 ,\quad
 \text{Banach-space fibers }\left(\cA_x:=\pi^{-1}x,\ \|\cdot\|_x\right)
 \end{equation*}
 of \cite[Definition II.13.4]{fd_bdl-1}, \cite[Definition 1.1]{dg_ban-bdl}, \cite[\S 3]{zbMATH03539905} and recalled below in \Cref{def:eqv.ban.bdl}. They \emph{can} be locally trivial in the sense of \cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \emph{continuous} in the sense that the globally-defined norm
 \begin{equation}\label{eq:glob.nrm}
 \cA
 \xrightarrow{\quad\|\cdot\|\quad}
 \bR_{\ge 0}
 ,\quad
 \|\cdot\||_{\cA_x}
 :=
 \|\cdot\|_x
 \end{equation}
 is continuous (\emph{(F)} Banach bundles in \cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).

\begin{definition}\label{def:eqv.ban.bdl}
 For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
 \begin{itemize}[wide]
 \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

The common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.
\begin{enumerate}[(a),wide]
\item\label{item:motiv.brnch} The \emph{non-commutative branched covers} introduced in \cite[Deﬁnition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\le A$ which satisfying convenient finiteness conditions (being of \emph{finite index} \cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \cite{bg_cx-exp,2409.17807v1,2409.03531v2}.

It is useful, in computing the dimension invariants attached by \cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \cite[Deﬁnition 2.3]{cpt_dim}) to finite-group actions on $\bS^{2n-1}_{\theta}$, to extend the $\bC^q$-fibered restriction of $\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \Cref{th:prlng.homog.abdls} affords this.
\end{enumerate}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

\begin{definition}\label{def:eqv.ban.bdl}
  For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
  \begin{itemize}[wide]
  \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

  \item with the \emph{fibers} $\cE_x:=\pi^{-1}x$, $x\in X$ carrying Banach-space structures compatible with the $\bU$-actions (which thus restrict to jointly-continuous representations $\bU\times \cE_x\to \cE_x$);

  \item so that scalar multiplication and addition are continuous;

  \item and the norm on $\cE$ is continuous (for \emph{continuous} or \emph{(F)} Banach bundles) or upper semicontinuous (for \emph{(H)} Banach bundles).
  \end{itemize}
  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\cE_x$ carrying those richer structures instead.
\end{definition}

\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

\begin{abstract}
 Consider a continuous bundle $\mathcal{E}\to X$ of Banach/Hilbert spaces or Banach/$C^*$-algebras over a paracompact base space, equivariant for a compact Lie group $\mathbb{U}$ operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from $\mathbb{U}$-invariant closed subsets of $X$ to closed invariant neighborhoods thereof (provided the fibers are semisimple in the Banach-algebra variant). This extends a number of results in the literature (due to Fell for non-equivariant local extensibility around a single point for $C^*$-algebras and the author for semisimple Banach algebras). The proofs are based in part on auxiliary results on (a) the extensibility of equivariant compact-Lie-group principal bundles locally around invariant closed subsets of paracompact spaces, as a consequence of equivariant-bundle classifying spaces being absolute neighborhood extensors in the relevant setting and (b) an equivariant-bundle version of Johnson's approximability of almost-multiplicative maps from finite-dimensional semisimple Banach algebras with Banach morphisms.
\end{abstract}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

The multiplicative counterpart to \Cref{th:prlng.homog.bdls} is as follows.

\begin{theorem}\label{th:prlng.homog.abdls}
 \begin{enumerate}[(1),wide]
 \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:
\begin{enumerate}[(a),wide]
\item\label{item:intro.equiv.bdls} There is, first, the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces. This forms the object of \Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \emph{sections} from closed subspaces: \cite[Theorem I.5.10]{kar_k_1978}, \cite[Th\'eor\`eme 3.3.1]{god_faisc_1958}, \cite[Proposition 1.1]{seg}, etc.

\item whereupon we can average against the Haar probability measure of $\bU$ to obtain a $\bU$-equivariant morphism $\cG\to \cE$ agreeing with the original embedding over $Z$ (much as in \cite[Proposition 1.1]{seg}, for instance).
 \end{itemize}
 Such an extension will still be an embedding
 \begin{equation*}
 \cG|_W
 \lhook\joinrel\xrightarrow{\quad \iota\quad}
 \cE|_W
 ,\quad
 \text{closed neighborhood }W\supseteq Z,
 \end{equation*}
 providing the desired \emph{Banach} bundle. In the more rigid setup of Hilbert bundles, an additional step will replace $\iota$ with the isometric component $\iota_{\mathrm{iso}}$ in the \emph{polar decomposition} \cite[\S I.5.2.2]{blk} $\iota=\iota_{\mathrm{iso}}|\iota|$.
\end{th:prlng.homog.bdls}

\begin{remarks}\label{res:gen.triv}
 \begin{enumerate}[(1),wide]
 \item For $\bU=\{1\}$ and \emph{trivial} (rather than locally trivial) $\cF$ \Cref{th:prlng.homog.bdls} reflects the fact (e.g. \cite[Corollary 2.10]{laz_bb-sel} or \cite[p.15]{dg_ban-bdl}) that bounded sections of (even (H), rather than (F)) Banach bundles extend from closed subsets of paracompact base spaces:
 \begin{itemize}[wide]
 \item having selected $n$ sections witnessing triviality across all of $Z$;

\begin{enumerate}[label={},wide]
 \item\textbf{\Cref{item:th:prlng.homog.abdls:ban}} In first instance, \Cref{pr:princ.bdl.ext} applied the automorphism group $\bG$ of some finite-dimensional $C^*$-algebra, i.e. a semidirect product of the form
 \begin{equation*}
 \left(\prod_{i=1}^k PU(n_i)\right)\rtimes \left(\text{finite group}\right),
 \end{equation*}
 will extend $\cB$ over a closed neighborhood (so we may as well assume globally) to a locally-trivial, $B$-fibered bundle$_{\bU}$ $\cM\xrightarrowdbl{\hspace{0pt}}X$ with $\cM|_A\cong \cB$. Precisely as in the proof of \Cref{th:prlng.homog.bdls}, we have a $\bU$-equivariant embedding
 \begin{equation*}
 \cM|_{W}
 \lhook\joinrel\xrightarrow{\quad\iota\quad}
 \cA|_W
 ,\quad
 \text{some closed neighborhood }W\supseteq Z
 \end{equation*}
 of \emph{vector} bundles, (unital and) multiplicative over $Z$. As a matter of convenience, a slight perturbation
 \begin{equation*}
 \iota
 \xmapsto{\quad}
 \bigg(
 \cM_x\ni a
 \xmapsto{\quad}
 \iota(a)+1_{\cA_x}-\iota\left(1_{\cM_x}\right)
 \in \cA_x
 \bigg)
 ,\quad
 x\in W
 \end{equation*}
 over sufficiently small $W$ followed by averaging over $\bU$ will further ensure $\iota$ is unital throughout $W$ (observe that this process will not have altered $\iota$ over the original smaller subspace $Z\subseteq X$).

\begin{proposition}\label{pr:princ.bdl.ext}
  Consider
  \begin{itemize}[wide]
  \item a compact Lie group $\bU$ acting via $\alpha:\bU\circlearrowright \bG$ on another;

  \item a closed $\bU$-equivariant embedding $Z\subseteq X$ into a paracompact $\bU$-space;

  \item and a principal $\bG$-bundle$_{\alpha}$ $\cP\xrightarrowdbl{\pi}Z$.
  \end{itemize}
  For some closed $\bU$-neighborhood $W\supseteq Z$ there is a principal $\bG$-bundle$_{\alpha}$ $\wt{\cP}\xrightarrowdbl{\hspace{0pt}}W$ with $\wt{\cP}|_Z\cong \cP$.
\end{proposition}

\begin{theorem}\label{th:prlng.homog.bdls}
  \begin{enumerate}[(1),wide]
  \item Let $\bU$ be a compact Lie group, $\cE\xrightarrowdbl{\hspace{0pt}} X$ a continuous Banach bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cF\le \cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\bU}$.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial $n$-homogeneous subbundle$_{\bU}$ $\wt{\cF}\xrightarrowdbl{\hspace{0pt}} W$ of $\cE|_W$;

    \item with $\wt{\cF}|_Z=\cF$.
    \end{itemize}

  \item The analogous result holds also for continuous Hilbert bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

The multiplicative counterpart to \Cref{th:prlng.homog.bdls} is as follows.

\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

\Cref{th:prlng.homog.abdls} generalizes \cite[Theorem 3.1]{fell_struct} (in the $C^*$ setting) and \cite[Proposition 2.8]{2409.03531v2} (for both $C^*$ and Banach-algebra bundles) by allowing equivariance and extensibility locally around arbitrary ($\bU$-invariant) closed subsets rather than single points.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:
\begin{enumerate}[(a),wide]
\item\label{item:intro.equiv.bdls} There is, first, the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces. This forms the object of \Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \emph{sections} from closed subspaces: \cite[Theorem I.5.10]{kar_k_1978}, \cite[Th\'eor\`eme 3.3.1]{god_faisc_1958}, \cite[Proposition 1.1]{seg}, etc.

\item\label{item:intro.amnm} Secondly, in order to leverage the ``purely-linear'' \Cref{th:prlng.homog.bdls} into the ``multiplicative'' \Cref{th:prlng.homog.abdls}, we employ an equivariant bundle version of Johnson's principle \cite[Theorem 3.1, Corollary 3.2, Theorem 7.2]{john_approx} that almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are close to morphisms in the apposite category. This is one instance of what has since come to be termed \emph{Hyers-Ulam-Rassias stability} \cite[Introduction]{jung_hur-stab} and the family-wide, equivariant adaptation of Johnson's arguments can be regarded as fitting in that broad framework.
\end{enumerate}

In reference to item \Cref{item:intro.equiv.bdls} above it becomes topical to contrast the various topologies one typically equips the total space of the \emph{(Milnor-model) universal bundle} $E\bG$ \cite[Definition 7.2.7]{hjjm_bdle} of a(n always Hausdorff) topological group. \Cref{se:colim.top} revolves around that issue:
\begin{itemize}[wide]
\item \Cref{th:disc.cone} observes that the principal action $E\bG\times \bG\to E\bG$ is discontinuous for the colimit topology on $E\bG$ whenever, say, $\bG\times -$ fails to preserve the topological quotient
  \begin{equation*}
    \bG\times [0,1]
    \xrightarrowdbl{\quad}
    \text{\emph{cone} }
    \cC \bG:=\bG\times [0,1]\big/\bG\times \{0\}.
  \end{equation*}
  Indeed, such groups fail to act continuously even on the truncated total space
  \begin{equation*}
    E_1\bG=\bG*\bG
    \underset{\text{closed}}{\subseteq}
    E\bG,
  \end{equation*}
  with `*' denoting the \emph{topological join} \cite[p.9]{hatch_at} recalled in \Cref{eq:x.ast.y}. As simple a group as $(\bQ,+)$ will exhibit this behavior: \Cref{pr:q.discont.act.join}.

\item while \Cref{th:lb.cc} notes that by contrast, a group with \emph{countably-compact} power $\bG^n$ always acts continuously on its colimit-topologized truncated universal space $E_n \bG=\bG^{*(n+1)}$.
\end{itemize}

\subsection*{Acknowledgments}

I am grateful for illuminating comments and suggestions from B. Badzioch, M. Megrelishvili and M. Ramachandran.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} are described as relying on two technical ingredients:
\begin{enumerate}[(a),wide]
\item the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces (\Cref{pr:princ.bdl.ext}), in a way "reminiscent" of extensibility results for sections from closed subspaces;
\item to pass from the "purely-linear" \Cref{th:prlng.homog.bdls} to the "multiplicative" \Cref{th:prlng.homog.abdls}, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are "close to morphisms" in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.
\end{enumerate}

The proofs of To prove the main theorem, and of the following theorem are described as relying on two technical ingredients:
\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}
\begin{enumerate}[(a),wide]
\item the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces.
We first prove the following proposition.
\begin{proposition}\label{pr:princ.bdl.ext}
  Consider
  \begin{itemize}[wide]
  \item a compact Lie group $\bU$ acting via $\alpha:\bU\circlearrowright \bG$ on another;

  \item a closed $\bU$-equivariant embedding $Z\subseteq X$ into a paracompact $\bU$-space;

  \item and a principal $\bG$-bundle$_{\alpha}$ $\cP\xrightarrowdbl{\pi}Z$.
  \end{itemize}
  For some closed $\bU$-neighborhood $W\supseteq Z$ there is a principal $\bG$-bundle$_{\alpha}$ $\wt{\cP}\xrightarrowdbl{\hspace{0pt}}W$ with $\wt{\cP}|_Z\cong \cP$.
\end{proposition}
\item to pass from the ``purely-linear'' To prove the main theorem, to the ``multiplicative'' theorem above, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are ``close to morphisms'' in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.
\end{enumerate},

Question: Let \(\mathbf U\) be a compact Lie group. A continuous \(\mathbf U\)-equivariant Banach bundle over a space \(X\) is a Banach bundle \(\pi:\mathcal E\to X\) equipped with continuous \(\mathbf U\)-actions on \(\mathcal E\) and \(X\) such that \(\pi\) is \(\mathbf U\)-equivariant and the norm on \(\mathcal E\) is continuous. A closed \(\mathbf U\)-subset \(Z\subseteq X\) means a closed \(\mathbf U\)-invariant subset. A locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) means a \(\mathbf U\)-equivariant locally trivial subbundle whose fibers all have dimension \(n\). Under these assumptions, which existence statement holds for Banach bundles and also for Hilbert bundles?

If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).

Choices:
B. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a \(\mathbf U\)-invariant open neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).
C. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\).
D. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is an \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).
E. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exists a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) such that for every continuous Banach bundle\(_{\mathbf U}\) structure on \(\mathcal E|_W\) extending the given one on \(\mathcal E|_Z\), there is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) with \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).

preservation of n-homogeneous fiber dimension and Hilbert-bundle analogue

initial local triviality of the subbundle on Z

existential quantifier for the ambient restriction and extension neighborhood


\label{thm0}
Let $d\geq 2$ be an integer and let $K\subset \mathbb{R}^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\in K$, $\Psi-\wtan(F,x)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$.

\begin{ex}\label{ex1}[A set of Hausdorff dimension $1$ which attains every pseudotangent along a Lipschitz curve]

For natural numbers $k\geq 2$, let $C_k\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\log(k)/\log(k+1)$. Then by self-similarity, $\dim_A(C_k)=\log(k)/\log(k+1)$ as well. In $\R^d$, for $d\geq 2$, let
\[K:=\{{\bf 0}\}\cup\left(\bigcup_{k\geq 2} \{0\}^{d-2}\times \{1/k\}\times C_k\right).\]
Then $K\subset \R^d$ is compact, $\dimh(K)=1$, $\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\times\{0\}$. Then for all $x\in K$, $\Psi-\wtan (F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.

\end{ex}

\label{prelim}
\begin{definition}
Given a set $K\subset \R^d$, we say that a map $f:[0,1]\to \R^d$ or its image $F=f([0,1])$ is a \emph{Lipschitz capture of} $K$ if $f$ is a Lipschitz function and $K\

\begin{ex}\label{ex2}[A Lipschitz curve with non-unique pseudotangents at every point]

Let $\{q_j\}_{j\in\N}$ be an enumeration of $\Q\cap [0,1]$, let $K_0:=[0,1]\times\{0\}\subset \R^2$, and let \[K_1:=K_0\cup\left(\bigcup_{j\in\N}\{q_j\}\times [0,\frac{1}{4(j+1)^2}]\right).\]
We proceed by recursion.

For $n\geq 1$, given finite-length continua $K_{n-1}\subset K_n$, satisfying that the components of $K_n\setminus K_{n-1}$ are either all parallel to the $x$-axis or all parallel to the $y$-axis, and such that $K_{n-1}\subset \overline{K_n\setminus K_{n-1}}$, we construct $K_{n+1}$ as follows. Let $\{(x_j^n,y_j^n)\}_{j\in\N}$ be a countable dense subset of $K_n\setminus K_{n-1}$. If the components of $K_n\setminus K_{n-1}$ are all parallel to the $x$-axis, then let
\[K_{n+1}:=K_n\cup\left(\bigcup_{j\in\N} \{x_j^n\}\times[y_j^n,y_j^n+\frac{1}{(j+1)^2(n+1)^2}]\right),\]
and if the components of $K_n\setminus K_{n-1}$ are instead all parallel to the $y$-axis, then let
\[K_{n+1}:=K_n\cup \left(\bigcup_{j\in\N}[x_j^n,x_j^n+\frac{1}{(j+1)^2(n+1)^2}]\times\{y_n^j\}\right).\]

The Hausdorff limit $K_\infty$ of these sets $K_n$ exists and by  \cite[Theorem 4.4]{AO} and by Go{\l}ab's Semicontinuity Theorem, it is a continuum of finite length, and so a Lipschitz curve. It is straightforward to see that since points with a unique tangent line are dense in $K_\infty$ and since points admitting a tangent which is not a line are also dense in $K_\infty$, every point of $K_\infty$ admits at least two distinct pseudotangents.
\end{ex}

\begin{theorem}\label{thm0}
Let $d\geq 2$ be an integer and let $K\subset \R^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\in K$, $\Psi-\wtan(F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.
\end{theorem}

Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\mathbb{R}^m\to \mathbb{R}^n$ is Lipschitz, then $f$ is differentiable at $\mathscr{L}^m$-almost every point $x\in \mathbb{R}^m$, where $\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\mathbb{R}^m$ (see for example \cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\subset \mathbb{R}^m$ such that for every Lipschitz function $f:\mathbb{R}^m\to \mathbb{R}^n$, there exists a point $x\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\subset\mathbb{R}^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \cite{PS15} for the proof and for a more complete history of this problem.

In addition to the classical version of Rademacher's Theorem, there are many notions of ``tangents to sets" which also admit a version of this theorem. If $\Gamma\subset \mathbb{R}^d$ is a Lipschitz curve, meaning that it is the image $f([0,1])$ of a Lipschitz map $f:[0,1]\to\mathbb{R}^d$, then $\Gamma$ admits a tangent line at $\mathcal{H}^1$-almost all of its points. This statement holds for approximate tangent planes (see \cite[Theorem 15.19]{Mattila}), for approximate tangent cones (see \cite[Theorem 3.8]{falc}), and most recently for tangents in the sense of Badger and Lewis \cite[Definition 3.1]{BL} (see \cite[Theorem 1.1]{QSS}). Here and throughout this article, $\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. Furthermore, with Vellis we proved in \cite[Appendix A]{QSS} that in every $\mathbb{R}^d$ with $d\geq 2$, there exists a Lipschitz curve $H\subset \mathbb{R}^d$ containing the origin ${\bf 0}$ such that every possible tangent is attained for $H$ at ${\bf 0}$. The goal of this paper is to expand on the work in this direction with the following result.

\maketitle
\section{Introduction}
Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\R^m\to \R^n$ is Lipschitz, then $f$ is differentiable at $\mathscr{L}^m$-almost every point $x\in \R^m$, where $\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\R^m$ (see for example \cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\subset \R^m$ such that for every Lipschitz function $f:\R^m\to \R^n$, there exists a point $x\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\subset\R^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \cite{PS15} for the proof and for a more complete history of this problem.

Here and throughout, we use the notation $\wtan (X,x)$ to denote the set of \emph{tangents} to the set $X$ at the point $x$, $\Psi-\wtan (X,x)$ to denote the set of \emph{pseudotangents} to the set $X$ at the point $x$, and $\mathfrak{C}_U(\R^d;{\bf 0})$ to denote the collection of all closed subsets of $\R^d$ which contain the origin and have only unbounded components, see Section \ref{prelim} for the details of these sets. In Proposition \ref{prop:alltan}, we prove that as long as $X\subset \R^d$ is a nondegenerate continuum, it holds that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\R^d;{\bf 0})$ for each point $x\in X$. Note that in Theorem \ref{thm0}, we obtain a result in terms of \emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \ref{ex2} we show that there is a nondegenerate Lipschitz curve where \emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\mathcal{H}^1$-almost everywhere, admit a unique \emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes".
\begin{question}\label{q1}
For each integer $d\geq 2$, does there exist a Lipschitz curve $\Gamma\subset \R^d$ such that for every point $x\in\Gamma$, $\Psi-\wtan(\Gamma,x)=\mathfrak{C}_U(\R^d;{\bf 0})$?
\end{question}

Our strategy for proving Theorem \ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\{z_k\}_{k\in\N}\subset G\setminus K$ for which for all $k\in\N$, $\wtan(G,z_k)=\mathfrak{C}_U(\R^d;{\bf 0})$ and such that $K\subset \overline{\{z_k\}}_{k\in\N}$. We then prove as a consequence that all of the points $z\in K$, $\Psi-\wtan(G,z)=\mathfrak{C}_U(\R^d;{\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\overline{\{z_k\}}_{k\in\N}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.

\begin{rem}\label{tanrem}
Note the following two facts about tangents and pseudotangents, the proofs of which are left to the interested reader.
\begin{enumerate}
\item For any non-empty closed set $K$ and any point $x\in K$, $\wtan(K,x)\subset\Psi-\wtan (K,x)$.
\item If $(x_i)_{i\in\N}\subset K$ converges to a point $x\in K$ and if $T\in\wtan(K,x_i)$ for each $i$, then $T\in\Psi-\wtan(K,x)$ as well.
\end{enumerate}
\end{rem}
Throughout this article, $H$ will denote the set of the same name as in \cite[Appendix A]{QSS}, to which we refer the reader for the details of its construction. The critical properties of $H$ for this article are that $H$ is a finite-length continuum, so by \cite[Theorem 4.4]{AO} it is a Lipschitz curve, which contains ${\bf 0}$, is contained in $[-1,1]^d$, and for which $\wtan(H,{\bf 0})=\mathfrak{C}_U(\R^d;{\bf 0})$.
\section{Results}\label{results}

Further, note that the following result holds, with proof exactly the same as the proof of \cite[Lemma 2.5]{QSS}, up to replacing instances of $x$ with $x_n$ instead (to move from tangents to pseudotangents). Structurally the proofs are identical and all of the arguments still hold \emph{mutatis mutandis}.
\begin{proposition}\label{prop:alltan}
Let $d\geq 2$ be an integer and let $X\subset \R^d$ be a nondegenerate continuum. Then for every point $x\in X$, we have that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\R^d;{\bf 0})$.
\end{proposition}
This means that the pseudotangent collection given in Theorem \ref{thm0} is maximal at all of the points of $K$. The proof of Theorem \ref{thm0} follows after several lemmata about the geometry of Lipschitz captures of the set $K$. Fix, for the remainder of this section, a compact, uniformly disconnected, perfect set $K\subset\R^d$ which admits a Lipschitz capture $G_0=g_0([0,1])$. By uniform disconnectedness, let $\lambda>0$ such that for every pair of distinct points $x,y\in K$, if $\{p_0,p_1,\dots,p_n\}\subset K$ with $p_0=x$ and $p_n=y$, then there exists some $i\in\{0,1,\dots,n-1\}$ for which $|p_{i+1}-p_i|>\lambda|x-y|$.

\begin{lemma}\label{lem2}
For every $x\in K$, there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that
\begin{equation}\label{eqn2}
\Psi-\wtan(F,x)=\mathfrak{C}_U(\R^d;{\bf 0}).
\end{equation}
Moreover, if $G=g([0,1])$ is any Lipschitz capture of $K$ and $r,\delta>0$ are any positive numbers, then we can find a Lipschitz capture $F=f([0,1])$ of $K$ satisfying (\ref{eqn2}) and further satisfying $F\setminus B(x,r)=G\setminus B(x,r)$ and $\mathcal{H}^1(F)-\mathcal{H}^1(G)< \delta$.
\end{lemma}
\begin{proof}
Fix a point $x\in K$ and let $(y_n)_{n\in\N}\subset K\setminus\{x\}$ be a sequence of (distinct) points with $\lim_{n\to\infty} y_n=x$ and with $\sum_{n\in\N}|x-y_n|<\infty$. Let $G=g([0,1])$ be a Lipschitz capture of $K$. For each $n\in\N$, note that $\dist(g^{-1}(\{x\}),g^{-1}(\{y_n\}))=:\eta_n>0$ with $\eta_n\to 0$ as $n\to\infty$, so let $a_n\in g^{-1}(\{x\})$ and $b_n\in g^{-1}(\{y_n\})$ realize $\eta_n=|a_n-b_n|$. Then by Lemma \ref{lem1}, for each $n\in\N$, there exists a value $\zeta_n\in(\min(\{a_n,b_n\}),\max(\{a_n,b_n\}))$ with $\dist(g(\zeta_n),K)> \frac{1}{4}\lambda|x-y_n|$ and such that $g(\zeta_n)$ is a point of $\mathcal{H}^1$ density $1$ in $G$.

First, observe that the Hausdorff limit $\lim_{n\to\infty} G_n=:G$ exists, $K\subset G$, and $G$ is a continuum of finite length by Go{\l}ab's Semicontinuity Theorem, so it is a Lipschitz capture of $K$ by \cite[Theorem 4.4]{AO} (see this same paper for discussion of Go{\l}ab's Semicontinuity Theorem as well). We now claim that for every $n\in\N$, $\Psi-\wtan (G,x_n)=\mathfrak{C}_U(\R^d;{\bf 0})$.

For natural numbers $k\geq 2$, let $C_k\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\log(k)/\log(k+1)$. Then by self-similarity, $\dim_A(C_k)=\log(k)/\log(k+1)$ as well. In $\R^d$, for $d\geq 2$, let
\[K:=\{{\bf 0}\}\cup\left(\bigcup_{k\geq 2} \{0\}^{d-2}\times \{1/k\}\times C_k\right).\]
Then $K\subset \R^d$ is compact, $\dimh(K)=1$, $\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\times\{0\}$. Then for all $x\in K$, $\Psi-\wtan (F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.

Here and throughout, we use the notation $\wtan (X,x)$ to denote the set of \emph{tangents} to the set $X$ at the point $x$, $\Psi-\wtan (X,x)$ to denote the set of \emph{pseudotangents} to the set $X$ at the point $x$, and $\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ to denote the collection of all closed subsets of $\mathbb{R}^d$ which contain the origin and have only unbounded components, see Section \ref{prelim} for the details of these sets. In Proposition \ref{prop:alltan}, we prove that as long as $X\subset \mathbb{R}^d$ is a nondegenerate continuum, it holds that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ for each point $x\in X$. Note that in Theorem \ref{thm0}, we obtain a result in terms of \emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \ref{ex2} we show that there is a nondegenerate Lipschitz curve where \emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\mathcal{H}^1$-almost everywhere, admit a unique \emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes".
\begin{question}\label{q1}
For each integer $d\geq 2$, does there exist a Lipschitz curve $\Gamma\subset \mathbb{R}^d$ such that for every point $x\in\Gamma$, $\Psi-\wtan(\Gamma,x)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$?
\end{question}

Our strategy for proving Theorem \ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ for which for all $k\in\mathbb{N}$, $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. We then prove as a consequence that all of the points $z\in K$, $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.

\subsection{Acknowledgments}
We would like to thank Jeremy Tyson for many helpful conversations giving this project direction, as well as for his commentary on an early draft. Additionally, we thank Vyron Vellis for suggesting Example \ref{ex1} and its inclusion in the final manuscript.

The post-theorem introduction gives the following strategy to prove Theorem~\ref{thm0}. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ such that for all $k$, one has $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$, and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. Then show that for every point $z$ in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$ (in particular for all $z\in K$), it follows that $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; the argument "does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure."

The post-theorem introduction gives the following strategy to prove the main theorem. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ such that for all $k$, one has $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$, and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. Then show that for every point $z$ in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$ (in particular for all $z\in K$), it follows that $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; the argument "does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure."

Question: Let \(d\ge 2\) be an integer, and let \(K\subset \mathbb{R}^d\) be compact and uniformly disconnected, meaning that there exists \(\lambda>0\) such that for every distinct \(x,y\in K\) and every finite chain \(\{p_0,p_1,\dots,p_n\}\subset K\) with \(p_0=x\) and \(p_n=y\), there is some \(i\in\{0,\dots,n-1\}\) for which \(|p_{i+1}-p_i|>\lambda |x-y|\). Assume also that \(K\) admits a Lipschitz capture, i.e. there exists a Lipschitz map \(f:[0,1]\to \mathbb{R}^d\) whose image contains \(K\). For a set \(X\subset \mathbb{R}^d\) and a point \(x\in X\), let \(\Psi\!-\mathrm{tan}(X,x)\) denote the set of pseudotangents to \(X\) at \(x\) (pointed blow-up limits with base points tending to \(x\)), and let \(\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\) be the collection of all closed subsets of \(\mathbb{R}^d\) that contain the origin and whose connected components are all unbounded. Under these assumptions, which existence statement holds?

There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).

Choices:
B. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in F\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).
C. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), one has \(\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\subseteq \Psi\!-\mathrm{tan}(F,x)\).
D. For every Lipschitz capture \(F=f([0,1])\) of \(K\) and every \(x\in K\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).
E. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), \(\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).