Π-trivial map
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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 | + | 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
Tex syntax errorbe a basepoint of . The set of homotopy classes of paths from
Tex syntax errorto is non-canonically isomophic to . An isomorphism is defined by a choice of path to represent the identity element:
Tex syntax error
Tex syntax error. Define an equivalence relation on this set by saying
Tex syntax errorto modulo . A choice of lift defines a bijection of sets
Tex syntax errorto . Conversely given a choice of class
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
Tex syntax errorto 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