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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2010 • Question 17

Question

Math 6510 • Differentiable Manifolds Spring 2010 • Question 17

Prove that a closed orientable surface \(S_g\) of genus \(g\ge 1\) is not homotopy equivalent to the wedge \(X\vee Y\) of two finite cell complexes both of which have nontrivial \(H_1(\cdot ;\mathbb {Z})\).

Raw LaTeX

Prove that a closed orientable surface \(S_g\) of genus \(g\ge1\) is not
homotopy equivalent to the wedge \(X\vee Y\) of two finite cell complexes both of
which have nontrivial \(H_1(\cdot;\Z)\).

Topic Tags

CW Complexes and Cell Attachments Homotopy Equivalence, Retracts, and Contractibility Homotopy and Degree on Manifolds Surface Topology and Surface Groups