4.II.21H

Algebraic Topology | Part II, 2006

Fix a point pp in the torus S1×S1S^{1} \times S^{1}. Let GG be the group of homeomorphisms ff from the torus S1×S1S^{1} \times S^{1} to itself such that f(p)=pf(p)=p. Determine a non-trivial homomorphism ϕ\phi from GG to the group GL(2,Z)\operatorname{GL}(2, \mathbb{Z}).

[The group GL(2,Z)\mathrm{GL}(2, \mathbb{Z}) consists of 2×22 \times 2 matrices with integer coefficients that have an inverse which also has integer coefficients.]

Establish whether each ff in the kernel of ϕ\phi is homotopic to the identity map.

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