Euler characteristic as surgery obstruction (Ex)

Latest revision as of 14:19, 6 January 2019

Consider a map $<wikitex>; Consider a map $f \colon M \to X$ from a closed $n$-dimensional manifold $M$ to a finite $CW$-complex $X$. Suppose that it can be converted by a finite sequence of surgery steps to a homotopy equivalence $f' \colon M' \to X$. Show that then $\chi(M) - \chi(X) \equiv 0 \mod 2$. </wikitex> [[Category:Exercises]] [[Category:Exercises without solution]]f \colon M \to X$ from a closed $n$-dimensional manifold $M$ to a finite $CW$-complex $X$. Suppose that it can be converted by a finite sequence of surgery steps to a homotopy equivalence $f' \colon M' \to X$. Show that then $\chi(M) - \chi(X) \equiv 0 \mod 2$.