Π-trivial map
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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.
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.
2 Definition
Tex syntax errorbe an
![m](/images/math/f/5/2/f52ba22baf75438bb1b02f476954c023.png)
![(\widetilde{M},\pi,w)](/images/math/0/c/1/0c1539ba36c32c4d6f6e4920967e4a21.png)
![\pi](/images/math/1/2/2/12240904cbf29d712b1aad3e405e04e9.png)
![f:N^n\to M^m](/images/math/6/9/7/697231d0e1b7682c309a42d3bf8b57ae.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\widetilde{f}:N \to \widetilde{M}](/images/math/f/c/d/fcd29866d8d481f764adea162c2534ba.png)
![\displaystyle \xymatrix{ \pi_1(N) \ar[r]^-{f_*} & \pi_1(M) \ar[r] & \pi }](/images/math/a/f/6/af62534d631fe62ab4fd71ba1af6a259.png)
is trivial.
3 Properties
![f:N\to M](/images/math/4/d/6/4d68520934f9523da856dce28d4d2871.png)
![\widetilde{M}](/images/math/a/c/4/ac49ca39301be0e9cc576dc7efc5f6e6.png)
![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![\widetilde{M}](/images/math/a/c/4/ac49ca39301be0e9cc576dc7efc5f6e6.png)
![\displaystyle f^*\widetilde{M} \cong N\times \pi.](/images/math/7/3/a/73a1bac2416135374351ad9b56e5e5a5.png)
![\widetilde{f}:N \to \widetilde{M}](/images/math/f/c/d/fcd29866d8d481f764adea162c2534ba.png)
![N\times \{1\} \subset N \times \pi](/images/math/7/b/6/7b699bc219f7c85a64d3c73c28d0f881.png)
![\widetilde{f}:\widetilde{N}:=N\times \pi \to \widetilde{M}](/images/math/1/8/b/18bfbdb807af78adca1b16108435fb48.png)
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