Thickenings
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# a basepoint $m \in \partial M \subset M$ and an orientation of $TM_m$, | # a basepoint $m \in \partial M \subset M$ and an orientation of $TM_m$, | ||
# a simple homotopy equivalence $\phi : K \to M$. | # 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 | + | 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{ | \xymatrix{ | ||
− | M_0 \ar[ | + | M_0 \ar[dr]^{\phi_0} \ar[0,2]^{f} & & M_1 \ar[dl]_{\phi_1} \\ |
− | K | + | & K} |
$$ | $$ | ||
{{endthm}} | {{endthm}} | ||
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</wikitex> | </wikitex> | ||
Revision as of 14:01, 27 November 2010
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
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- the map is an isomorphism,
-
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is homotopy equivalent to .
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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 or denotes respectively the topological, piecewise linear or smooth categories.
Definition 0.1 [Wall1966a, Section 1]. Let be a finite connected CW complex. An -dimensional -thickening of consists of
- a compact -dimensional -manifold
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with connected boundary such that - a basepoint and an orientation of ,
- a simple homotopy equivalence .
Two thickenings and are isomorphic if there is a -isomorphism preserving and the orientation of and such that is simple homotopic to . In particular there is a simple homotopy commutative diagram:
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