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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 9

Let \(f,g:K\to L\) be two chain maps (\(f_n,g_n:K_n\to L_n\)). Define what it means for \(f\) and \(g\) to be chain homotopic. Prove that if \(f\) and \(g\) are chain homotopic, then they induce the same maps in homology, \(f_*=g_*:H_n(K)\to H_n(L)\).

Raw LaTeX

Let \(f,g:K\to L\) be two chain maps
(\(f_n,g_n:K_n\to L_n\)). Define what it means for \(f\) and \(g\) to be chain homotopic.
Prove that if \(f\) and \(g\) are chain homotopic, then they induce the same maps in
homology, \(f_*=g_*:H_n(K)\to H_n(L)\).

Topic Tags

Chain Complexes and Chain Homotopy Homology with Coefficients Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds