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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2013 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2013 • Question 8

Let \(S_g\) denote the closed oriented surface of genus \(g\). Show that if \(h<g\) then every map \(S_h\to S_g\) has degree 0. Does every such map have to be null-homotopic?

Raw LaTeX

Let \(S_g\) denote the closed oriented surface of genus \(g\). Show that if
\(h<g\) then every map \(S_h\to S_g\) has degree 0. Does every such map have to be
null-homotopic?

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups