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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2006 • Question 16

Question

Math 6510 • Differentiable Manifolds Fall 2006 • Question 16

Show that if the closed orientable surface \(M_g\) of genus \(g\) retracts onto a graph \(X\subset M_g\), then \(H_1(X)\) has rank at most \(g\). You may use the following algebraic fact: a nonsingular skew-symmetric bilinear pairing over the rationals \(\mathbb {Q}\), of the form \(\mathbb {Q}^n\times \mathbb {Q}^n\to \mathbb {Q}\), cannot be identically 0 when restricted to \(V\times V\) for any \(\mathbb {Q}\)-linear subspace \(V\subset \mathbb {Q}^n\) of dimension \(>n/2\).

Raw LaTeX

Show that if the closed orientable surface \(M_g\) of genus \(g\) retracts onto a graph
\(X\subset M_g\), then \(H_1(X)\) has rank at most \(g\). You may use the following algebraic
fact: a nonsingular skew-symmetric bilinear pairing over the rationals \(\mathbb{Q}\), of
the form \(\mathbb{Q}^n\times\mathbb{Q}^n\to\mathbb{Q}\), cannot be identically 0 when
restricted to \(V\times V\) for any \(\mathbb{Q}\)-linear subspace
\(V\subset\mathbb{Q}^n\) of dimension \(>n/2\).

Topic Tags

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