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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2005 • Question 14

Question

Math 6510 • Differentiable Manifolds Winter 2005 • Question 14

Let \[ f:S^1\times S^1\to S^1\times S^1 \] be map of the torus to itself. Since \(H_1(S^1\times S^1)=\mathbb {Z}\oplus \mathbb {Z}\), after choosing generators for \(H_1(S^1\times S^1)\) we can represent \[ f_*:H_1(S^1\times S^1)\to H_1(S^1\times S^1) \] as a \(2\times 2\) matrix \(A\) with integer entries. If \(f\) does not have fixed points what are the possibilities for the trace and determinant of \(A\) and the degree of \(f\)? For each possible triple give an example of a fixed point free map.

Raw LaTeX

Let
\[
  f:S^1\times S^1\to S^1\times S^1
\]
be map of the torus to itself. Since
\(H_1(S^1\times S^1)=\mathbb{Z}\oplus\mathbb{Z}\), after choosing generators for
\(H_1(S^1\times S^1)\) we can represent
\[
  f_*:H_1(S^1\times S^1)\to H_1(S^1\times S^1)
\]
as a \(2\times2\) matrix \(A\) with integer entries. If \(f\) does not have fixed points what
are the possibilities for the trace and determinant of \(A\) and the degree of \(f\)? For
each possible triple give an example of a fixed point free map.

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Fundamental Groups Homotopy and Degree on Manifolds Surface Topology and Surface Groups