Π-trivial map
(Finished the rest of relating lifts to paths) |
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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. | ||
</wikitex> | </wikitex> | ||
Revision as of 17:42, 21 May 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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 .
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 .
3 Properties
4 Lifts and paths - two alternative perspectives
Tex syntax error
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.
5 Examples
...
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