Π-trivial map

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	A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$.		A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$.
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		+	== Lifts and paths - two alternative perspectives ==
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		+	Since $\pi$ is the group of deck transformations of $\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$ with the group structure determined by the action of $\pi$ once an identity element lift $\widetilde{f}_{id}$ has been chosen. In this way the choice of lift that is included as part of the data of a $\pi$-trivial map can be thought of as a choice of isomorphism $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f} = g\widetilde{f}_{id}.\end{array}$$
	</wikitex>		</wikitex>

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Revision as of 10:54, 2 May 2013

This page has not been refereed. The information given here might be incomplete or provisional.

Contents 1 Introduction 2 Definition 3 Properties 4 Lifts and paths - two alternative perspectives 5 Examples 6 References

1 Introduction

This page is based on [Ranicki2002]. A map ${{Stub}} == Introduction == <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 $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. 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. </wikitex> == Definition == <wikitex>; 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 $$\xymatrix{ \pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi }$$ is trivial. </wikitex> == Properties == <wikitex>; A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$. </wikitex> == Lifts and paths - two alternative perspectives == <wikitex>; Since $\pi$ is the group of deck transformations of $\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$ with the group structure determined by the action of $\pi$ once an identity element lift $\widetilde{f}_{id}$ has been chosen. In this way the choice of lift that is included as part of the data of a $\pi$-trivial map can be thought of as a choice of isomorphism $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f} = g\widetilde{f}_{id}.\end{array}$$ </wikitex> == Examples == <wikitex>; ... </wikitex> ==References== {{#RefList:}} [[Category:Definitions]]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 with covering map $p:\widetilde{M} \to M$. If $f$ factors through

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$\widetilde{M}$ 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. [Hatcher2002, Proposition 1.33]) a map $f:N\to M$ can be lifted to

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$\widetilde{M}$ 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.

2 Definition

Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an 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

is trivial.

3 Properties

A map $f:N\to M$ that factors through

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$\widetilde{M}$ must map all of $N$ to the same sheet of

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$\widetilde{M}$, hence the pullback satisfies Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$.

4 Lifts and paths - two alternative perspectives

Since $\pi$ is the group of deck transformations of

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$\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$ with the group structure determined by the action of $\pi$ once an identity element lift $\widetilde{f}_{id}$ has been chosen. In this way the choice of lift that is included as part of the data of a $\pi$-trivial map can be thought of as a choice of isomorphism

5 Examples

...

6 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