Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 4 / 21H / Solution

Identify the pointed torus with ${\mathbb{R}}^2/{\mathbb{Z}}^2$ with $p$ at the origin. Its fundamental group is ${\mathbb{Z}}^2$, generated by the two coordinate loops. A homeomorphism fixing $p$ induces an automorphism of this group, givingFunctoriality makes it a homomorphism. It is nontrivial, indeed onto: every integer invertible matrix defines a pointed torus homeomorphism by $x\mapsto Ax$ modulo the Euclidean lattice.

If $f_{\ast }=I$, lift $f$ to a continuous $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ with $F(0)=0$. Its deck-transformation law is $F(x+n)=F(x)+n$ for every $n\in {\mathbb{Z}}^2$, because the induced map on coordinate loops is identity. Thenis well defined: adding $n$ changes the bracketed expression by exactly $n$. It is a continuous pointed homotopy from $f$ to identity. Hence every kernel element is homotopic to the identity, although the intermediate maps need not be homeomorphisms.