A new approach to spines is available via mass transport theory and Kantorovich duality. This is developed in my PhD thesis.

The idea is elementary: consider the retract $x\mapsto x/|x|$ from the closed unit ball $B:=\{x \in \mathbb{R}^{N} | ~~ |x| \leq 1\}$ to its boundary sphere $\partial B$. The retract has locus-of-discontinuity $Z=\{pt\}$ equal to a point, namely $x=0$. Observe that the inclusion $\{pt\} \hookrightarrow B$ is a homotopy-equivalence. Claim: this is general principle which follows from Kantorovich duality and the optimal transport theory.

For example, let $S$ be closed hyperbolic surface with metric $d$, and $C\hookrightarrow S$ an embedded Cantor set. Let $X:=S-C$ be the Cantor-punctured surface, and let $\sigma$ be the Hausdorff measure on $X$. Similarly let $\tau$ be the Hausdorff measure of $C$ viewed as a subset of $(S,d)$. Now consider the function $c: X \times C \to (0,\infty)$ defined by the rule $$c(x,y_0):= [\int_C d(x,y)^{-2} d\tau(y) ] - \frac{1}{2} d(x,y_0)^{-2}.$$ We view $c(x,y_0)$ as the cost of transporting a unit mass from the source $x\in X$ to target $y_0\in C$. If $\int_X \sigma > \int_C \tau$, then there exist semicoupling measures $\pi$ on $X\times C$ with the property $$proj_X \# \pi \leq \sigma, ~~~\text{and}~~~~proj_C \# \pi = \tau.$$ In otherwords, $\pi$ is a transference plan from the abundant source $\sigma$ to the prescribed target $tau$. (Such measures are called "Semicouplings"). It is standard result of optimal transport that there exists a unique $c$-optimal semicoupling $\pi_*$ which minimizes the total cost $$c[\pi]:=\int_{X\times C} c(x,y) d\pi(x,y).$$

Now imagine we rescale the target measure $\tau\mapsto \lambda \tau$ for scalar $\lambda>0$. If $\lambda \int_C \tau$ is sufficiently close to $\int_X \sigma$, then the $c$-optimal semicoupling $\pi_*$ will have a "locus-of-discontinuity" $Z \hookrightarrow X$ such that $Z$ is a strong-deformation retract of $X$ and $Z$ will be codimension-one (i.e., the "singularity" is the spine).

More specifically, the $c$-optimal semicouplings $\pi_*$ are characterized by the existence of a $c$-concave potentials $\psi: C \to \mathbb{R}$ satisfying $(\psi^c)^c=\psi$. This is Kantorovich's duality theory. The $c$-optimal transport has the form $x\mapsto \partial^c \psi^c (x)$ for every $x\in X$. Here $\partial^c \psi^c(x)$ is a subset of $C$, namely the $c$-subdifferential of $\psi^c$ at $x\in X$. The "locus-of-discontinuity" is more precisely described as the set of $x\in X$ where $\# \partial^c \psi^c (x) \geq 2$, i.e. where the $c$-convex potential $\psi^c$ is not uniquely differentiable. The locus-of-discontinuity $Z$, where $\psi^c$ is finite and not uniquely differentiable, is a closed lipschitz subvariety of $X$. And Kantorovich duality shows $Z \hookrightarrow X$ is a deformation-retract. The existence of this retract is probably not obvious, unless you are well-studied in the mass-transport theory...

But all the details are in my thesis, including applications to spines for the Teichmueller spaces and symmetric spaces of arithmetic-groups. I'd be happy to share the details, since my supervisor has absolutely zero interest in topological applications, and is quite plainly indifferent to algebraic topology.

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