Π-trivial map

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1 Introduction

This page is based on [Ranicki2002]. A map $f:N^n\to M^m$${{Stub}} == Introduction == <wikitex>; This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented Cover|oriented cover]] with covering map $p:\widetilde{M} \to M$. If $f$ factors through $\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a '''choice''' of lift $\widetilde{f}$ is required in order to represent a homology class. By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N)\to \pi_1(M) \to \pi$ is trivial for $q:\pi_1(M)\to \pi_1(M)/\pi_1(\widetilde{M}) = \pi$ the quotient map. </wikitex> == Definition == <wikitex>; Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented Cover|oriented cover]]. A '''$\pi$-trivial map''' $f:N^n\to M^m$ is a map from an oriented manifold $N$, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$, such that the composite $$\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.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$. </wikitex> == Examples == <wikitex>; ... </wikitex> == References == {{#RefList:}} [[Category:Definitions]]f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$$f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover with covering map $p:\widetilde{M} \to M$$p:\widetilde{M} \to M$. If $f$$f$ factors through $\widetilde{M}$$\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$$f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$$f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$$\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a choice of lift $\widetilde{f}$$\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$$f:N\to M$ can be lifted to $\widetilde{M}$$\widetilde{M}$ if and only if $f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$$f_*(\pi_1(N)) \subset p_*(\pi_1(\widetilde{M}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N)\to \pi_1(M) \to \pi$$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$$q:\pi_1(M)\to \pi_1(M)/\pi_1(\widetilde{M}) = \pi$ the quotient map.

2 Definition

Let $M$$M$ be an $m$$m$-dimensional manifold and let $(\widetilde{M},\pi,w)$$(\widetilde{M},\pi,w)$ be an oriented cover. A $\pi$$\pi$-trivial map $f:N^n\to M^m$$f:N^n\to M^m$ is a map from an oriented manifold $N$$N$, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$$\widetilde{f}:N \to \widetilde{M}$, such that the composite

$\displaystyle \xymatrix{ \pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi }$

is trivial.

3 Properties

A map $f:N\to M$$f:N\to M$ that factors through $\widetilde{M}$$\widetilde{M}$ must map all of $N$$N$ to the same sheet of $\widetilde{M}$$\widetilde{M}$, hence the pullback satisfies

$\displaystyle f^*\widetilde{M} \cong N\times \pi.$ $$\displaystyle f^*\widetilde{M} \cong N\times \pi.$$

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

4 Examples

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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