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Mapping class groups of $h$-cobordant manifolds

Samuel Muñoz-Echániz

TL;DR

The paper shows that the mapping class group is not an $h$-cobordism invariant in high dimensions by constructing $h$-cobordant manifolds with mapping class groups of different cardinalities, distinguished by $3$-adic valuations. It develops moduli spaces of $h$-block and $s$-block bundles and builds an algebraic model $F^{\mathsf{alg}}_\bullet(A)$ to realize the homology with a $C_2$-action, connecting to $H(-)_{hC_2}$ via a Dold–Kan framework. A lens-space based $h$-cobordism $W:L\to M$ is constructed to force differences in the inertial and block mapping class data, and the Rothenberg exact sequence alongside Weiss–Williams theory is used to relate these differences to $L$-theory and stable concordance phenomena. The results provide concrete obstructions to invariance under $h$-cobordism and reveal intricate interactions between $h$- and $s$-block moduli and diffeomorphism groups in high dimensions.

Abstract

We prove that the mapping class group is not an $h$-cobordism invariant of high-dimensional manifolds by exhibiting $h$-cobordant manifolds whose mapping class groups have different cardinalities. In order to do so, we introduce a moduli space of "$h$-block" bundles and understand its difference with the moduli space of ordinary block bundles.

Mapping class groups of $h$-cobordant manifolds

TL;DR

The paper shows that the mapping class group is not an -cobordism invariant in high dimensions by constructing -cobordant manifolds with mapping class groups of different cardinalities, distinguished by -adic valuations. It develops moduli spaces of -block and -block bundles and builds an algebraic model to realize the homology with a -action, connecting to via a Dold–Kan framework. A lens-space based -cobordism is constructed to force differences in the inertial and block mapping class data, and the Rothenberg exact sequence alongside Weiss–Williams theory is used to relate these differences to -theory and stable concordance phenomena. The results provide concrete obstructions to invariance under -cobordism and reveal intricate interactions between - and -block moduli and diffeomorphism groups in high dimensions.

Abstract

We prove that the mapping class group is not an -cobordism invariant of high-dimensional manifolds by exhibiting -cobordant manifolds whose mapping class groups have different cardinalities. In order to do so, we introduce a moduli space of "-block" bundles and understand its difference with the moduli space of ordinary block bundles.
Paper Structure (20 sections, 27 theorems, 129 equations, 4 figures)

This paper contains 20 sections, 27 theorems, 129 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Moduli spaces of h- and s-block bundles
  4. Notation and recollections
  5. Whitehead Torsion
  6. The s-cobordism theorem
  7. The block moduli spaces of manifolds
  8. A simplicial model for H(-)hC2
  9. The simplicial abelian group $F^{\mathsf{alg}}_\bullet(A)$
  10. Proof of Theorem \ref{['ActualThmC']}
  11. Proof of Theorem \ref{['ThmB']}
  12. Relation to the Rothenberg exact sequence
  13. Proof of Theorem A(i)
  14. A general strategy
  15. The candidate W:L->M
  16. ...and 5 more sections

Key Result

Theorem A

In each dimension $d=12k-1\geq 0$, there exist $d$-manifolds $M^d$ (see Theorem ThmAiequivalent) $h$-cobordant to the lens space $L=L_7^{12k-1}(r_1:\dots: r_{6k})$, where such that

Figures (4)

  • Figure 1: Illustration of $W_\#V$ and $P:=V\cup_{L\times\{0\}}W$ when $n=2$.
  • Figure 2: $h$-cobordisms rel boundary $D(W)\space \raisebox{-1pt}{$\overset{h}{\leadsto}$}\space L\times I$ and $D(\overline W)\space \raisebox{-1pt}{$\overset{h}{\leadsto}$}\space M\times I$.
  • Figure 3: The map $\mathcal{R}_\bullet$ with $p=2$ and $\dim M=0$.
  • Figure 4: Geometric Eilenberg swindle.

Theorems & Definitions (78)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem B
  • Theorem 2.1: $s$-Cobordism Theorem rel boundary
  • Definition 3.1
  • Example 3.3
  • Definition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 68 more