Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable moduli spaces of odd-dimensional manifold triads

João Lobo Fernandes

TL;DR

This work extends the stable moduli-space program of Galatius and Randal-Williams to odd-dimensional manifold triads with fixed vertical boundary, introducing a boundary-connected-sum stabilization by $V_g$ and a Moore–Postnikov $n$-factorization framework for tangential data. The main result identifies the stabilized moduli space of $ abla$-structures on a triad with the homotopy orbit of the infinite-loop Thom spectrum $\mathbf{MT}\Theta_N$, via an acyclic map onto a path component of $\Omega^{\infty}\mathbf{MT}\Theta_N$. A Kreck-type stable classification for odd triads follows, linking stable diffeomorphism types to stable normal $n$-types and Euler characteristics. The paper develops a comprehensive cobordism-category toolkit with boundary, refined surgery arguments, and a group-completion strategy, culminating in a flexible general theorem for $\\Theta$-structures and a suite of examples, including reductions to existing even-dimensional results. Overall, the results provide a robust odd-dimensional analogue of the GRW program, with broad implications for the homotopy type of diffeomorphism classifying spaces and stable diffeomorphism classifications in the presence of boundary data.

Abstract

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is $1$-connected. Stabilization is performed by taking boundary connected sum with $S^n \times D^{n+1}$. This is an analog of earlier work of Galatius and Randal-Williams for even-dimensional manifolds with fixed boundary, and it extends a previous result by Botvinnik and Perlmutter. As a byproduct, we obtain an analog for odd-dimensional triads of Kreck's stable diffeomorphism classification of even-dimensional manifolds.

Stable moduli spaces of odd-dimensional manifold triads

TL;DR

This work extends the stable moduli-space program of Galatius and Randal-Williams to odd-dimensional manifold triads with fixed vertical boundary, introducing a boundary-connected-sum stabilization by and a Moore–Postnikov -factorization framework for tangential data. The main result identifies the stabilized moduli space of -structures on a triad with the homotopy orbit of the infinite-loop Thom spectrum , via an acyclic map onto a path component of . A Kreck-type stable classification for odd triads follows, linking stable diffeomorphism types to stable normal -types and Euler characteristics. The paper develops a comprehensive cobordism-category toolkit with boundary, refined surgery arguments, and a group-completion strategy, culminating in a flexible general theorem for -structures and a suite of examples, including reductions to existing even-dimensional results. Overall, the results provide a robust odd-dimensional analogue of the GRW program, with broad implications for the homotopy type of diffeomorphism classifying spaces and stable diffeomorphism classifications in the presence of boundary data.

Abstract

We establish a homotopy-theoretic description of the homology of stable moduli spaces of -dimensional manifold triads with fixed , whenever and is -connected. Stabilization is performed by taking boundary connected sum with . This is an analog of earlier work of Galatius and Randal-Williams for even-dimensional manifolds with fixed boundary, and it extends a previous result by Botvinnik and Perlmutter. As a byproduct, we obtain an analog for odd-dimensional triads of Kreck's stable diffeomorphism classification of even-dimensional manifolds.
Paper Structure (27 sections, 48 theorems, 62 equations, 2 figures)

This paper contains 27 sections, 48 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction and statement of results.
  2. Pairs of spaces and manifolds.
  3. Strong connectivity.
  4. Connectivity for pairs.
  5. Triad homotopy groups.
  6. Pairs of vector bundles.
  7. Pairs of manifolds and triads.
  8. Handle decompositions for pairs of manifolds.
  9. Strong geometrical connectivity.
  10. Cobordism categories with boundary.
  11. Surgery on morphisms.
  12. Stable stability.
  13. Proof of the closure properties.
  14. Stable homology and group completion.
  15. The group completion argument.
  16. ...and 12 more sections

Key Result

Theorem 1

Let $(N,\partial^h N,\partial^v N)$ be a compact smooth $(2n+1)$-dimensional manifold triad where $N$ is connected, $(N,\partial^h N)$ is $1$-connected, $\partial^{hv} N\neq \emptyset$, and $n\geq 3$. For a Moore--Postnikov $n$-factorization $\tau_N=\Theta_N\circ l_N$, the map is acyclic onto the path component it hits.

Figures (2)

  • Figure 1: This is a triad cobordism $(W, \partial^h W,\partial_1 W, \partial^v W)$ from $(M,\partial^h M,\partial^v M)$ to $(N,\partial^h N,\partial^v N).$
  • Figure 2: This is a triad cobordism $W:M\leadsto N$ obtained from the trivial triad cobordism $(M\times [0,1],\partial^h M\times [0,1], \partial^v M\times [0,1], M\times \{0,1\})$ by attaching a left $k$-handle to $M\times \{1\}$. On the other hand, it is obtained from the trivial cobordism $N\times [0,1]$ by attaching a right $(d-k)$-handle to $N\times \{0\}$ (compare with \ref{['dual handle']}). We also depict the attaching map $f$, belt sphere $f'$ (see \ref{['elementary cobordisms']}), the core $(D^k_+,\partial_0D^k_+)$ and cocore $D^{d-k}.$

Theorems & Definitions (116)

  • Theorem 1
  • Remark
  • Theorem 2: \ref{['final with null']}
  • Corollary 3: Stable diffeomorphism classification
  • Definition 2.1.1
  • Remark 2.1.2
  • Lemma 2.1.3: 2 out of 3
  • proof
  • Lemma 2.1.4: Pushouts
  • proof
  • ...and 106 more