[Paper Review] Transversal homotopy theory

Jonathan Woolf|ArXiv.org|Oct 17, 2009

Homotopy and Cohomology in Algebraic Topology11 references21 citations

TL;DR

This paper develops transversal homotopy theory for Whitney stratified manifolds by restricting homotopies to only those maps that are transversal to each stratum, thereby preserving topological interactions with strata that would otherwise be lost in standard homotopy theory. The key contribution is a construction of transversal homotopy monoids and groupoids that categorically model $k$-tuply monoidal $n$-groupoids, with the Pontrjagin–Thom construction providing a geometric interpretation as framed tangles.

ABSTRACT

Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants. We also assign `higher invariants', the transversal homotopy categories, to each Whitney stratified manifold. These have a rich structure; they are rigid monoidal categories for n>1 and ribbon categories for n>2. As an example we show that the transversal homotopy categories of a sphere, stratified by a point and its complement, are equivalent to categories of framed tangles.

Motivation & Objective

• To establish a homotopy theory for Whitney stratified manifolds that respects strata by restricting to transversal maps, rather than approximating to transversality.
• To generalize classical homotopy invariants like homotopy groups and fundamental groupoids to a transversal setting that retains information about stratum crossings.
• To construct transversal homotopy monoids and groupoids that model higher categorical structures such as $k$-tuply monoidal $n$-groupoids.
• To provide a geometric realization of these invariants via the Pontrjagin–Thom construction, identifying transversal homotopy classes with framed tangles.
• To show that collapse maps for framed submanifolds are well-defined up to homotopy through transversal maps, ensuring consistency of the construction.

Proposed method

• Define transversal homotopy monoids $\psi_n(X)$ as functors from the category of pointed Whitney stratified manifolds to the category of dagger monoids, using maps $[0,1]^n \to X$ transverse to all strata.
• Introduce the category $\mathbf{Whit}_\star$ of pointed Whitney stratified manifolds and stratified normal submersions to ensure transversality is preserved under composition.
• Construct fibrant replacements (fattened versions) of standard topological constructions like suspensions and Thom spaces to ensure they lie in $\mathbf{Whit}_\star$, preserving transversality.
• Use the Pontrjagin–Thom construction to define collapse maps $\kappa_W: (S^n, B^n) \to (S^k, \star)$ for framed codimension $k$ submanifolds $W$, ensuring transversality and framing compatibility.
• Establish uniqueness of collapse maps up to homotopy through transversal maps by constructing a smoothing of the collapse map and using isotopy invariance.
• Show that any transversal map $f: (S^n, B^n) \to (S^k, \star)$ is homotopic through transversal maps to a collapse map for $f^{-1}(p)$, via a homotopy fixing a neighborhood of $p$.

Experimental results

Research questions

• RQ1 Can a homotopy theory be developed for stratified spaces that preserves information about transverse intersections with strata, rather than allowing cancellation?
• RQ2 How do transversal homotopy monoids $\psi_n(X)$ generalize classical homotopy groups $\pi_n(X)$ in the context of Whitney stratified manifolds?
• RQ3 What is the geometric meaning of $\psi_n(S^k)$, the transversal homotopy monoid of a sphere stratified by a point and its complement?
• RQ4 Can the transversal homotopy groupoids $\Psi_{n,n+1}(X)$ be constructed to model higher categorical structures such as $k$-tuply monoidal $n$-categories?
• RQ5 Are collapse maps for framed submanifolds well-defined up to homotopy through transversal maps, ensuring consistency in the invariants?

Key findings

• The transversal homotopy monoid $\psi_n(X)$ is a functor from $\mathbf{Whit}_\star$ to the category of dagger monoids, generalizing the classical $\pi_n(X)$ for $n > 0$.
• For the stratified sphere $\mathbb{S}^k$ with a point and its complement, $\psi_n(\mathbb{S}^k)$ is isomorphic to the set of ambient isotopy classes of codimension $k$ framed submanifolds in $\mathbb{R}^n$, i.e., framed tangles.
• The Pontrjagin–Thom construction provides a geometric model of $\psi_n(\mathbb{S}^k)$ as the framed tangle category in $n$ dimensions, establishing a direct link between transversal homotopy and geometric topology.
• Collapse maps for framed submanifolds are unique up to homotopy through transversal maps, ensuring that the construction of $\psi_n(X)$ is well-defined and consistent.
• Any transversal map $f: (S^n, B^n) \to (S^k, \star)$ is homotopic through transversal maps to a collapse map for the preimage $f^{-1}(p)$, showing that all transversal maps are representable by such geometric data.
• The transversal homotopy groupoid $\Psi_{n,n+1}(X)$ generalizes the fundamental groupoid and models a $k$-tuply monoidal $n$-category when $X = \mathbb{S}^k$, supporting the Tangle Hypothesis in this context.