Thickenings

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1 Introduction

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

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

Definition 0.1 [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:

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

2 References

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