Π-trivial map

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1 Introduction

This page is based on [Ranicki2002]. A map $f:N^n\to M^m$${{Stub}} == Introduction == <wikitex>; This page is based on \cite{Ranicki2002}. A map $f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$. Let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]] with covering map $p:\widetilde{M} \to M$. If $f$ factors through $\widetilde{M}$ as $f= p\circ \widetilde{f}: N \to \widetilde{M}\to M$ then $f$ represents a homology class $\widetilde{f}_*[N]\in H_n(\widetilde{M})$. Note that a '''choice''' of lift $\widetilde{f}$ is required in order to represent a homology class. Let $b_1$ be a basepoint of $N$. By covering space theory (c.f. \cite{Hatcher2002|Proposition 1.33}) a map $f:N\to M$ can be lifted to $\widetilde{M}$ if and only if $f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$ is trivial for $q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$ the quotient map. The group $\pi$ is well-defined for any choice of lift $\widetilde{f(b_1)}$ since $p:\widetilde{M}\to M$ is a regular covering and changing the basepoint in $\widetilde{M}$ to a different lift corresponds to conjugating $\pi_1(\widetilde{M},\widetilde{f(b_1)})$ by some $g\in \pi_1(M,f(b_1))$. </wikitex> == Definition == <wikitex>; Let $M$ be an $m$-dimensional manifold and let $(\widetilde{M},\pi,w)$ be an [[Oriented cover|oriented cover]]. A '''$\pi$-trivial map''' $f:N^n\to M^m$ is a map from an oriented manifold $N$ with basepoint $b_1$ such that the composite $$\xymatrix{ \pi_1(N,b_1) \ar[r]^-{f_*} & \pi_1(M,f(b_1)) \ar[r] & \pi }$$ is trivial, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$. </wikitex> == Properties == <wikitex>; A map $f:N\to M$ that factors through $\widetilde{M}$ must map all of $N$ to the same sheet of $\widetilde{M}$, hence the pullback satisfies $$f^*\widetilde{M} \cong N\times \pi.$$ Choosing where to lift a single point determines a lift $\widetilde{f}:N \to \widetilde{M}$, which thought of as a map from $N\times \{1\} \subset N \times \pi$ extends equivariantly to a lift $\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}$. </wikitex> == Lifts and paths - two alternative perspectives == <wikitex>; Rather than taking a lift as part of the data for a $\pi$-trivial map we could instead take an equivalence class of paths in $M$ as is explained in this section. Since $\pi$ is the group of deck transformations of $\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$ with the group structure determined by the action of $\pi$ once a choice of lift $\widetilde{f}_{id}$ has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a $\pi$-trivial map can be thought of as a choice of isomorphism $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f}(b_1) = g\widetilde{f}_{id}(b_1).\end{array}$$ Let $b$ be a basepoint of $M$. The set of homotopy classes of paths from $b$ to $f(b_1)$ is non-canonically isomophic to $\pi_1(M,b)$. An isomorphism is defined by a choice of path $[w_{id}]$ to represent the identity element: $$\begin{array}{rcl} \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}& \to & \pi_1(M,b)\ [w] &\mapsto & [w_{id}^{-1}*w],\end{array}$$where $*$ denotes concatenation of paths and $w_{id}^{-1}$ is the path $w_{id}$ in reverse. Let $\widetilde{b}$ be a basepoint of $\widetilde{M}$ that is a lift of $b$. Define an equivalence relation $\sim$ on this set by saying $$[w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).$$ The above isomorphism given by choosing $[w_{id}]$ descends to give an isomorphism $$\{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,$$ where we use the same choice of path $[w_{id}]$ to identify $\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b}) $ with $\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$. Thus a choice of lift $\widetilde{f}:N\to \widetilde{M}$ corresponds to a choice of homotopy class of paths from $b$ to $f(b_1)$ modulo $\pi_1(\widetilde{M})$. A choice of lift $\widetilde{b}$ defines a bijection of sets $$\{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$ as follows. Given a choice of lift $\widetilde{f}$ choose any path $\widetilde{w}:I \to \widetilde{M}$ from $\widetilde{b}$ to $\widetilde{f}(b_1)$. Take the equivalence class of $p(\widetilde{w})$ which is a path in $M$ from $b$ to $f(b_1)$. Conversely given a choice of class $$[w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$ choose any representative $w:I \to M$. This lifts uniquely to a path $\widetilde{w}$ starting at $\widetilde{b}$. Define a lift $\widetilde{f}$ by setting $\widetilde{f}(b_1):= \widetilde{w}(1)$. Note this map is well-defined since different choices of representative $w$ may differ by elements of $\pi_1(\widetilde{M},\widetilde{b})$ but their lifts will still end at the same point. To sum up we have the following diagram of non-canonical isomorphisms and bijections $$\xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \ & \pi & }.$$ Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups. </wikitex> == Examples == <wikitex>; ... </wikitex> ==References== {{#RefList:}} [[Category:Definitions]]f:N^n\to M^m$ between manifolds represents a homology class $f_*[N] \in H_n(M)$$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.

Let $b_1$$b_1$ be a basepoint of $N$$N$. 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,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$$f_*(\pi_1(N,b_1)) \subset p_*(\pi_1(\widetilde{M},\widetilde{f(b_1)}))$, i.e. if and only if the composition $q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$$q\circ f_*:\pi_1(N,b_1)\to \pi_1(M,f(b_1)) \to \pi$ is trivial for $q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$$q:\pi_1(M,f(b_1))\to \pi:=\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})$ the quotient map.

The group $\pi$$\pi$ is well-defined for any choice of lift $\widetilde{f(b_1)}$$\widetilde{f(b_1)}$ since $p:\widetilde{M}\to M$$p:\widetilde{M}\to M$ is a regular covering and changing the basepoint in $\widetilde{M}$$\widetilde{M}$ to a different lift corresponds to conjugating $\pi_1(\widetilde{M},\widetilde{f(b_1)})$$\pi_1(\widetilde{M},\widetilde{f(b_1)})$ by some $g\in \pi_1(M,f(b_1))$$g\in \pi_1(M,f(b_1))$.

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$ with basepoint $b_1$$b_1$ such that the composite

$\displaystyle \xymatrix{ \pi_1(N,b_1) \ar[r]^-{f_*} & \pi_1(M,f(b_1)) \ar[r] & \pi }$

is trivial, together with a choice of lift $\widetilde{f}:N \to \widetilde{M}$$\widetilde{f}:N \to \widetilde{M}$.

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 Lifts and paths - two alternative perspectives

Rather than taking a lift as part of the data for a $\pi$$\pi$-trivial map we could instead take an equivalence class of paths in $M$$M$ as is explained in this section. Since $\pi$$\pi$ is the group of deck transformations of $\widetilde{M}$$\widetilde{M}$, the set of lifts $\{\widetilde{f}:N\to \widetilde{M}\}$$\{\widetilde{f}:N\to \widetilde{M}\}$ is non-canonically isomorphic to $\pi$$\pi$ with the group structure determined by the action of $\pi$$\pi$ once a choice of lift $\widetilde{f}_{id}$$\widetilde{f}_{id}$ has been chosen to represent the identity element. In this way the choice of lift that is included as part of the data of a $\pi$$\pi$-trivial map can be thought of as a choice of isomorphism

$\displaystyle \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f}(b_1) = g\widetilde{f}_{id}(b_1).\end{array}$ $$\displaystyle \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f}(b_1) = g\widetilde{f}_{id}(b_1).\end{array}$$

Let $b$$b$ be a basepoint of $M$$M$. The set of homotopy classes of paths from $b$$b$ to $f(b_1)$$f(b_1)$ is non-canonically isomophic to $\pi_1(M,b)$$\pi_1(M,b)$. An isomorphism is defined by a choice of path $[w_{id}]$$[w_{id}]$ to represent the identity element:

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$$\displaystyle \begin{array}{rcl} \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}& \to & \pi_1(M,b)\\ [w] &\mapsto & [w_{id}^{-1}*w],\end{array}$$

where $*$$*$ denotes concatenation of paths and $w_{id}^{-1}$$w_{id}^{-1}$ is the path $w_{id}$$w_{id}$ in reverse. Let $\widetilde{b}$$\widetilde{b}$ be a basepoint of $\widetilde{M}$$\widetilde{M}$ that is a lift of $b$$b$. Define an equivalence relation $\sim$$\sim$ on this set by saying

$\displaystyle [w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).$ $$\displaystyle [w]\sim[w^\prime] \iff [w^{-1}*w^\prime] \in p_*(\pi_1(\widetilde{M},\widetilde{b})).$$

The above isomorphism given by choosing $[w_{id}]$$[w_{id}]$ descends to give an isomorphism

$\displaystyle \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,$ $$\displaystyle \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \;\longrightarrow\; \pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})\cong \pi,$$

where we use the same choice of path $[w_{id}]$$[w_{id}]$ to identify $\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})$$\pi_1(M,b)/\pi_1(\widetilde{M},\widetilde{b})$ with $\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$$\pi_1(M,f(b_1))/\pi_1(\widetilde{M},\widetilde{f(b_1)})=\pi$. Thus a choice of lift $\widetilde{f}:N\to \widetilde{M}$$\widetilde{f}:N\to \widetilde{M}$ corresponds to a choice of homotopy class of paths from $b$$b$ to $f(b_1)$$f(b_1)$ modulo $\pi_1(\widetilde{M})$$\pi_1(\widetilde{M})$. A choice of lift $\widetilde{b}$$\widetilde{b}$ defines a bijection of sets

$\displaystyle \{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$ $$\displaystyle \{\widetilde{f}:N\to \widetilde{M}\} \longleftrightarrow \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$

as follows. Given a choice of lift $\widetilde{f}$$\widetilde{f}$ choose any path $\widetilde{w}:I \to \widetilde{M}$$\widetilde{w}:I \to \widetilde{M}$ from $\widetilde{b}$$\widetilde{b}$ to $\widetilde{f}(b_1)$$\widetilde{f}(b_1)$. Take the equivalence class of $p(\widetilde{w})$$p(\widetilde{w})$ which is a path in $M$$M$ from $b$$b$ to $f(b_1)$$f(b_1)$. Conversely given a choice of class

$\displaystyle [w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$ $$\displaystyle [w]\in \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim$$

choose any representative $w:I \to M$$w:I \to M$. This lifts uniquely to a path $\widetilde{w}$$\widetilde{w}$ starting at $\widetilde{b}$$\widetilde{b}$. Define a lift $\widetilde{f}$$\widetilde{f}$ by setting $\widetilde{f}(b_1):= \widetilde{w}(1)$$\widetilde{f}(b_1):= \widetilde{w}(1)$. Note this map is well-defined since different choices of representative $w$$w$ may differ by elements of $\pi_1(\widetilde{M},\widetilde{b})$$\pi_1(\widetilde{M},\widetilde{b})$ but their lifts will still end at the same point.

To sum up we have the following diagram of non-canonical isomorphisms and bijections

$\displaystyle \xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \\ & \pi & }.$

Each map is obtained by making a choice and any two choices uniquely determine the third with the diagram commuting, so with two choices made the horizontal bijection is in fact an isomorphism of groups.

5 Examples

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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