Thickenings

Revision as of 15:45, 27 November 2010

1 Introduction

Let ${{Stub}}== Introduction == <wikitex>; Let $K$ be a finite connected CW-complex of dimension $K$. For a given $n > k$ we would like to know if there is a compact manifold $M$ with boundary $\partial M$ such that: # the map $\pi_1(\partial M) \to \pi_1(M)$ is an isomorphism, # $M$ is homotopy equivalent to $K$. In this case we say that $M$ ''thickens'' $K$. If there is such a manifold $M$, we would like to know how many up to homeomorphism or diffeomorphism if $M$ is smooth. In \cite{Wall1966a} Wall introduced the notion of a ''thickening'', defined below, to investigate the questions raised above. This page summarizes the basis results concerning thickenings. Recall that $\Cat = \Top, \PL$ or $\Diff$ denotes respectively the topological, piecewise linear or smooth categories. {{beginthm|Definition|\cite{Wall1966a|Section 1}}} Let $K$ be a finite connected CW complex. An $n$-dimensional ''$\Cat$-thickening'' of $K$ consists of # a compact $n$-dimensional $\Cat$-manifold $M$ with connected boundary such that $\pi_1(\partial M) \cong \pi_1(M)$ # a basepoint $m \in \partial M \subset M$ and an orientation of $TM_m$, # a simple homotopy equivalence $\phi : K \to M$. Two thickenings $(M_0, \phi_0)$ and $(M_1, \phi_1)$ are ''isomorphic'' if there is a $\Cat$-isomorphism $f \colon M_0 \cong M_1$ preserving $m$ and the orientation of $TM_m$ and such that $f_0$ is simple homotopic to $\phi_1 \circ f$. In particular there is a simple homotopy commutative diagram: $$ \xymatrix{ M_0 \ar[dr]^{\phi_0} \ar[0,2]^{f} & & M_1 \ar[dl]_{\phi_1} \ & K} $$ {{endthm}} </wikitex> == Examples == == Classification theorems == == References == {{#RefList:}} [[Category:Theory]]K$ be a finite connected CW-complex of dimension $K$. For a given $n = k + q > k$ we would like to know if there is a compact manifold $M$ with boundary $\partial M$ such that:

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

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

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

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

Definition 0.1 [Wall1966a, Section 1]. Let $K$ be a finite connected CW complex. An $n$-dimensional $\Cat$-thickening of $K$ consists of

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

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

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

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

2 Constructions and examples

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

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

3 Invariants

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

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

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

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

where $f_{TM} : M \to B\Cat$ is the classifying map.

4 Classification

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

Theorem 3.1 \cite{} \cite{.} For all $n \geq k$, the stable classifying map gives rise to a set bijection

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

5 References

• [Wall1966a] C. T. C. Wall, Classification problems in differential topology. IV. Thickenings, Topology 5 (1966), 73–94. MR0192509 (33 #734) Zbl 0149.20501