# Thickenings

## 1 Introduction

Let $K$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}K$ be a finite connected CW-complex of dimension $k$$k$. For a given $n = k + q > k$$n = k + q > k$ we would like to know if there is a compact manifold $M$$M$ with boundary $\partial M$$\partial M$ such that:

1. the map $\pi_1(\partial M) \to \pi_1(M)$$\pi_1(\partial M) \to \pi_1(M)$ is an isomorphism,
2. $M$$M$ is homotopy equivalent to $K$$K$.

In this case we say that $M$$M$ thickens $K$$K$. If there is such a manifold $M$$M$, we would like to know how many up to homeomorphism or diffeomorphism if $M$$M$ is smooth.

In [Wall1966a] Wall introduced the notion of a thickening, defined below, to investigate the questions raised above. This page summarizes the basic results concerning thickenings.

Recall that $\Cat = \Top, \PL$$\Cat = \Top, \PL$ or $\Diff$$\Diff$ denotes respectively the topological, piecewise linear or smooth categories.

Definition 1.1 [Wall1966a, Section 1]. Let $K$$K$ be a finite connected CW complex. An $n$$n$-dimensional $\Cat$$\Cat$-thickening of $K$$K$ consists of

1. a compact $n$$n$-dimensional $\Cat$$\Cat$-manifold $M$$M$ with connected boundary such that $\pi_1(\partial M) \cong \pi_1(M)$$\pi_1(\partial M) \cong \pi_1(M)$
2. a basepoint $m \in \partial M \subset M$$m \in \partial M \subset M$ and an orientation of $TM_m$$TM_m$,
3. a simple homotopy equivalence $\phi : K \to M$$\phi : K \to M$.

Two thickenings $(M_0, \phi_0)$$(M_0, \phi_0)$ and $(M_1, \phi_1)$$(M_1, \phi_1)$ are isomorphic if there is a $\Cat$$\Cat$-isomorphism $f \colon M_0 \cong M_1$$f \colon M_0 \cong M_1$ preserving $m$$m$ and the orientation of $TM_m$$TM_m$ and such that $f_0$$f_0$ is simple homotopic to $\phi_1 \circ f$$\phi_1 \circ f$. In particular there is a simple homotopy commutative diagram:

$\displaystyle \xymatrix{ M_0 \ar[dr]^{\phi_0} \ar[0,2]^{f} & & M_1 \ar[dl]_{\phi_1} \\ & K}$

The set of isomorphism classes of $n$$n$-dimensional $\Cat$$\Cat$-thickenings over $K$$K$ is denoted

$\displaystyle \mathcal{T}^n(K) := \{ [\phi: K \simeq M ] \}.$

## 2 Constructions and examples

The simplest examples of thickenings come from $q$$q$-disc bundles with sections over manifolds, $q > 2$$q > 2$. Let $X$$X$ be a closed $\Cat$$\Cat$-manifold of dimension $k$$k$, let $M \to X$$M \to X$ be a bundle with fibre $D^q$$D^q$ and with section $s \colon M \to X$$s \colon M \to X$. Then

• the pair $(M, s)$$(M, s)$ is an $n$$n$-thickening of $X$$X$.
• the pair $(D^n, pt)$$(D^n, pt)$ is an $n$$n$-thickening of a point.

## 3 Invariants

An important invariant of a thickening $(M, \phi)$$(M, \phi)$ is the induced stable $\Cat$$\Cat$-bundle over $K$$K$:

$\displaystyle T(M, \phi) := \phi^* TM$

where $TM \to M$$TM \to M$ is the stable $\Cat$$\Cat$-tangent bundle of $M$$M$. Given that stable bundles over the space $K$$K$ are classified by maps to the classifying space $B\Cat$$B\Cat$ one equivalently thinks of

$\displaystyle T(M, \phi) = [f_{TM} \circ \phi]$

where $f_{TM} : M \to B\Cat$$f_{TM} : M \to B\Cat$ is the classifying map. Clearly the isomorphism class of the bundle $\phi^* TM \to K$$\phi^* TM \to K$ is an invariant of the thickening class $[M, \phi]$$[M, \phi]$. This if $Bun_\Cat(K)$$Bun_\Cat(K)$ denotes the set of isomorphism classes of stable $\Cat$$\Cat$-bundles over $K$$K$ we obtain a map

$\displaystyle \tau_K : \mathcal{T}^n(K) \to Bun_\Cat(K).$

## 4 Classification

An extremely useful classification theorem in manifold theory is the classification of stable thickenings where $n \geq 2k+1$$n \geq 2k+1$ originally due to Wall in the smooth catagory.

Theorem 4.1 [Wall1966a, Proposition 5.1] [Chazin1971, Theorem 1]. For all $n \geq 2k+1$$n \geq 2k+1$, the stable classifying map gives rise to a set bijection

$\displaystyle \tau_K : \mathcal{T}^n(K) \equiv [K, B\Cat], \quad [M, \phi] \mapsto T(M, \phi) = [\phi \circ f_{TM}].$