Tangential homotopy equivalences (Ex)

A homotopy equivalence $f \colon M \to N$$<wikitex>; A homotopy equivalence $f \colon M \to N$ of closed smooth $n$-manifolds is called ''stably tangential'' if $f^*(\tau_N) \cong \tau_M$; i.e. $f$ pulls back the stable tangent bundle of $N$ to $M$. {{beginthm|Example}} # Give an example of a stably tangential homotopy equivalence which is not homotopic to a diffeomorphism. # Show that if $f \colon M \to N$ is a stably tangential homotopy equivelance, then there is a diffeomorphism $$ F \colon D^{n+1} \times M \cong D^{n+1} \times N$$ such that $p_n \circ F|_{0 \times M}$ is homotopic to $f$, where $p_N \colon D^{n+1} \times N \to N$ is the projection. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]f \colon M \to N$ of closed smooth $n$$n$-manifolds is called stably tangential if $f^*(\tau_N) \cong \tau_M$$f^*(\tau_N) \cong \tau_M$; i.e. $f$$f$ pulls back the stable tangent bundle of $N$$N$ to $M$$M$.

Example 0.1.

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

$\displaystyle F \colon D^{n+1} \times M \cong D^{n+1} \times N$ $$\displaystyle F \colon D^{n+1} \times M \cong D^{n+1} \times N$$

such that $p_n \circ F|_{0 \times M}$$p_n \circ F|_{0 \times M}$ is homotopic to $f$$f$, where $p_N \colon D^{n+1} \times N \to N$$p_N \colon D^{n+1} \times N \to N$ is the projection.