Tangential homotopy equivalences (Ex)
From Manifold Atlas
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| − | A homotopy equivalence $f \colon M \to N$ of closed smooth $n$-manifolds is called ''tangential'' if $f^* | + | A homotopy equivalence $f \colon M \to N$ of closed smooth $n$-manifolds is called ''stably tangential'' if $f^*(\tau_N) \cong \tau_M$; i.e. $f$ pulls back the stable tangent bundle of $N$ to $M$. |
{{beginthm|Example}} | {{beginthm|Example}} | ||
| − | # Give an example of a tangential homotopy equivalence which is not homotopic to a diffeomorphism. | + | # Give an example of a stably tangential homotopy equivalence which is not homotopic to a diffeomorphism. |
| − | # Show that if $f \colon M \to N$ is a tangential homotopy equivelance, then there is a diffeomorphism | + | # Show that if $f \colon M \to N$ is a stably tangential homotopy equivelance, then there is a diffeomorphism |
| − | $$ F \colon D^n \times M \cong D^n \times N | + | $$ F \colon D^{n+1} \times M \cong D^{n+1} \times N$$ |
| + | such that $p_n \circ F|_{0 \times M}$ is homotopic to $f$, where $p_N \colon D^{n+1} \times N \to N$ is the projection. | ||
{{endthm}} | {{endthm}} | ||
</wikitex> | </wikitex> | ||
Latest revision as of 23:53, 25 August 2013
A homotopy equivalence 
of closed smooth 
-manifolds is called stably tangential if 
; i.e. 
pulls back the stable tangent bundle of 
to 
.
Example 0.1.
- Give an example of a stably tangential homotopy equivalence which is not homotopic to a diffeomorphism.
- Show that if is a stably tangential homotopy equivelance, then there is a diffeomorphism
such that 
is homotopic to , where 
is the projection.
