Tangential homotopy equivalences (Ex)
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A homotopy equivalence $f \colon M \to N$ of closed smooth $n$-manifolds is called ''tangential'' if $f^*TN \cong TM$; i.e. $f$ pulls back the tangent bundle of $N$ to $M$. | A homotopy equivalence $f \colon M \to N$ of closed smooth $n$-manifolds is called ''tangential'' if $f^*TN \cong TM$; i.e. $f$ pulls back the 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. |
# 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 tangential homotopy equivelance, then there is a diffeomorphism | ||
$$ F \colon D^n \times M \cong D^n \times N.$$ | $$ F \colon D^n \times M \cong D^n \times N.$$ |
Revision as of 23:50, 25 August 2013
A homotopy equivalence of closed smooth -manifolds is called tangential if ; i.e. pulls back the tangent bundle of to .
Example 0.1.
- Give an example of a tangential homotopy equivalence which is not homotopic to a diffeomorphism.
- Show that if is a tangential homotopy equivelance, then there is a diffeomorphism