Π-trivial map
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== Lifts and paths - two alternative perspectives == | == Lifts and paths - two alternative perspectives == | ||
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− | 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 $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g\;s.t.\; \widetilde{f}(b_1) = g\widetilde{f}_{id}(b_1).\end{array}$$ | + | 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 |
+ | $$ \begin{array}{rcl} \{\widetilde{f}:N\to \widetilde{M}\} & \stackrel{\simeq}{\longrightarrow} & \pi \\ \widetilde{f} & \mapsto & g \; \; \mathrm{s.t.}\; \widetilde{f}(b_1) = g\widetilde{f}_{\!\id}(b_1).\end{array}$$ | ||
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 $b$. | 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 $b$. | ||
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To sum up we have the following diagram of non-canonical isomorphisms and bijections | 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. | $$\xymatrix{ \{\widetilde{f}:N\to \widetilde{M}\} \ar@{<->}[rr]^-{\widetilde{b}} \ar[dr]_-{\widetilde{f}_{id}} && \{[w:I\to M]: w(0)=b,\,w(1)=f(b_1)\}/\sim \ar[dl]^-{[w_{id}]} \\ & \pi & }.$$ 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. | ||
+ | Since an oriented cover comes with a choice of lift $\widetilde{b}$ as '''part of the data''' a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a $\pi$-trivial map. | ||
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== Examples == | == Examples == | ||
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− | ... | + | Let $f:S^n\looparrowright M^{2n}$ be an immersion and let $\widetilde{M}$ be the universal cover of $M$, let $s\in S^n$ and $b\in M$ be basepoints. For $n>1$, $\pi_1(S^n)=0$ so $f$ lifts to $\widetilde{M}$. An immersion $f:S^n\looparrowright M^{2n}$ is a $\pi_1(M)$-trivial immersion as soon as a lift $\widetilde{f}:S^n \looparrowright \widetilde{M}$ has been prescribed or, alternatively, once a homotopy class of paths $w:I\to M$ from $b$ to $f(s)$ has been prescribed. A pair $(f:S^n\looparrowright M^{2n}, w: I\to M | w(0)=b \; \text{and} \; w(1)=f(s))$ is often called a pointed immersion in the literature (See, for example, \cite{Lück2001|Section 4.1}). |
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Latest revision as of 17:26, 16 June 2014
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
[edit] 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.
Let be a basepoint of . 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.
The group is well-defined for any choice of lift since is a regular covering and changing the basepoint in to a different lift corresponds to conjugating by some .
[edit] 2 Definition
Let be an -dimensional manifold and let be an oriented cover. A -trivial map is a map from an oriented manifold with basepoint such that the composite
is trivial, together with a choice of lift .
[edit] 3 Properties
[edit] 4 Lifts and paths - two alternative perspectives
Rather than taking a lift as part of the data for a -trivial map we could instead take an equivalence class of paths in as is explained in this section. Since is the group of deck transformations of , the set of lifts is non-canonically isomorphic to with the group structure determined by the action of once a choice of lift 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 -trivial map can be thought of as a choice of isomorphism
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To sum up we have the following diagram of non-canonical isomorphisms and bijections
Since an oriented cover comes with a choice of lift as part of the data a choice of identity lift corresponds to a choice of identity path, so it does not matter which we choose to include as part of the data for a -trivial map.
[edit] 5 Examples
Let be an immersion and let be the universal cover of , let and be basepoints. For , so lifts to . An immersion is a -trivial immersion as soon as a lift has been prescribed or, alternatively, once a homotopy class of paths from to has been prescribed. A pair is often called a pointed immersion in the literature (See, for example, [Lück2001, Section 4.1]).
[edit] 6 References
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Lück2001] W. Lück, A basic introduction to surgery theory, 9 (2001), 1–224. Available from the author's homepage. MR1937016 (2004a:57041) Zbl 1045.57020
- [Ranicki2002] A. Ranicki, Algebraic and geometric surgery, The Clarendon Press Oxford University Press, Oxford, 2002. MR2061749 (2005e:57075) Zbl 1003.57001