Π-trivial map
(Difference between revisions)
m (Change to emphasise that a choice of lift is required for a Pi-trivial map to represent homology in the cover and that this lift should be part of the data in being a Pi-trivial map) |
|||
Line 25: | Line 25: | ||
</wikitex> | </wikitex> | ||
− | == References == | + | ==References== |
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Revision as of 15:20, 19 April 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
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. 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.
2 Definition
Let be an -dimensional manifold and let be an oriented cover. A -trivial map is a map from an oriented manifold , together with a choice of lift , such that the composite
is trivial.
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 .
4 Examples
...
5 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