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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Fall 2020 • Question 4

Question

Math 6520 • Algebraic Topology Fall 2020 • Question 4

Recall that the mapping torus of a map \(f:X\to X\) is the quotient space \(T_f\) obtained from \(X\times [0,1]\) by identifying \((x,1)\) with \((f(x),0)\) for every \(x\in X\). Compute the homology groups of \(T_f\) when \(f:S^n\to S^n\) is a map of degree \(d\) and \(n > 1\).

Raw LaTeX

Recall that the mapping torus of a map \(f:X\to X\) is the quotient space
\(T_f\) obtained from \(X\times[0,1]\) by identifying \((x,1)\) with \((f(x),0)\) for every
\(x\in X\). Compute the homology groups of \(T_f\) when \(f:S^n\to S^n\) is a map
of degree \(d\) and \(n > 1\).

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Homology with Coefficients Mapping Tori Surface Topology and Surface Groups