Π-trivial map
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m (Change to emphasise that a choice of lift is required for a Pi-trivial map to represent homology in the cover and that this lift should be part of the data in being a Pi-trivial map) |
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
− | This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented Cover|oriented cover]] with covering map $p:\widetilde{M} \to M$. If $f$ factors through $\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. | + | This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented Cover|oriented cover]] with covering map $p:\widetilde{M} \to M$. If $f$ factors through $\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a '''choice''' of lift $\widetilde{f}$ is required in order to represent a homology class. |
By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N)\to \pi_1(M) \to \pi$ is trivial for $q:\pi_1(M)\to \pi_1(M)/\pi_1(\widetilde{M}) = \pi$ the quotient map. | By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N)\to \pi_1(M) \to \pi$ is trivial for $q:\pi_1(M)\to \pi_1(M)/\pi_1(\widetilde{M}) = \pi$ the quotient 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 Cover|oriented cover]]. A '''$\pi$-trivial map''' $f:N^n\to M^m$ is a map from an oriented manifold $N$ such that the composite | + | 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 12:51, 19 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