Tangential homotopy equivalences (Ex)

Revision as of 23:51, 25 August 2013

A homotopy equivalence $<wikitex>; 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}} # 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 $$ F \colon D^n \times M \cong D^n \times N.$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]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 $M$.

Example 0.1.

1. Give an example of a tangential homotopy equivalence which is not homotopic to a diffeomorphism.
2. Show that if $f \colon M \to N$ is a tangential homotopy equivelance, then there is a diffeomorphism

$$\displaystyle F \colon D^n \times M \cong D^n \times N$$

such that $p_n \circ F|_{0 \times M}$ is homotopic to $f$, where $p_N \colon D^n \times N \to N$ is the projection.

References