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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== 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 $ | + | 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
Tex syntax erroras 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
Tex syntax errorif 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
Tex syntax errorto 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
Tex syntax errormust map all of to the same sheet of
Tex syntax error, hence the pullback satisfies
Choosing where to lift a single point determines a lift , which thought of as a map from extends equivariantly to a lift .
[edit] 4 Lifts and paths - two alternative perspectives
Tex syntax error, 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
Let be a basepoint of . The set of homotopy classes of paths from to is non-canonically isomophic to . An isomorphism is defined by a choice of path to represent the identity element:
Tex syntax error
Tex syntax errorthat is a lift of . Define an equivalence relation on this set by saying
The above isomorphism given by choosing descends to give an isomorphism
where we use the same choice of path to identify with .
Thus a choice of lift corresponds to a choice of homotopy class of paths from to modulo . A choice of lift defines a bijection of sets
as follows. Given a choice of lift choose any path from to . Take the equivalence class of which is a path in from to . Conversely given a choice of class
choose any representative . This lifts uniquely to a path starting at . Define a lift by setting . Note this map is well-defined since different choices of representative may differ by elements of but their lifts will still end at the same point.
To sum up we have the following diagram of non-canonical isomorphisms and bijections
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 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
Tex syntax errorbe the universal cover of , let and be basepoints. For , so lifts to
Tex syntax error. 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