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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 15

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 15

Suppose \((C_k,\partial _k)\) is a free chain complex. Suppose that it is acyclic, i.e. the homology groups \(H_k(C)=0\) for all \(k\). Show that \(C\) is contractible, that is, the identity map and the zero map \(\iota _k,\zeta _k:C_k\to C_k\) are chain homotopic.

Raw LaTeX

Suppose \((C_k,\partial_k)\) is a free chain complex. Suppose that it is
\emph{acyclic}, i.e. the homology groups \(H_k(C)=0\) for all \(k\). Show that \(C\) is
\emph{contractible}, that is, the identity map and the zero map
\(\iota_k,\zeta_k:C_k\to C_k\) are chain homotopic.

Topic Tags

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