Π-trivial map
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== Definition == | == Definition == | ||
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− | Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented | + | Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]]. A '''$\pi$-trivial map''' $f:N^n\to M^m$ is a map from an oriented manifold $N$, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$, such that the composite |
$$\xymatrix{ | $$\xymatrix{ | ||
\pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi | \pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi |
Revision as of 15:50, 26 April 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
This page is based on [Ranicki2002]. A map between manifolds represents a homology class . Let be an oriented cover with covering map . If factors through as then represents a homology class . Note that a choice of lift is required in order to represent a homology class.
By covering space theory (c.f. [Hatcher2002, Proposition 1.33]) a map can be lifted to if and only if , i.e. if and only if the composition is trivial for the quotient map.
2 Definition
Let be an -dimensional manifold and let be an oriented cover. A -trivial map is a map from an oriented manifold , together with a choice of lift , such that the composite
is trivial.
3 Properties
Choosing where to lift a single point determines a lift , which thought of as a map from extends equivariantly to a lift .
4 Examples
...
5 References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001