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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2018 • Question 7

Question

Math 6510 • Differentiable Manifolds Spring 2018 • Question 7

Suppose \(X\) and \(Y\) are cell complexes and that \(A\subseteq X\) is a subcomplex. Let \(f:A\to Y\) be continuous and let \(Y\sqcup _f X\) be \(Y\) with \(X\) attached along \(A\) via \(f\). Prove that \(Y\sqcup _f X\) is homotopy equivalent to a cell complex.

Raw LaTeX

Suppose \(X\) and \(Y\) are cell complexes and that \(A\subseteq X\) is a
subcomplex. Let \(f:A\to Y\) be continuous and let \(Y\sqcup_f X\) be \(Y\) with \(X\)
attached along \(A\) via \(f\). Prove that \(Y\sqcup_f X\) is homotopy equivalent to a
cell complex.

Topic Tags

CW Complexes and Cell Attachments Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Simplicial and Delta Complexes