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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2020 • Question 4

Question

Math 6510 • Differentiable Manifolds Summer 2020 • Question 4

Let \(M\) and \(N\) be a smooth orientable manifolds. Show that the maps \(i_a:M\to \mathbb {R}\times M\) given by \(i_a(x)=(a,x)\) and \(\pi :\mathbb {R}\times M\to M\) given by \(\pi (t,x)=x\) induce inverse maps in de Rham Cohomology. Conclude that if smooth maps \(f:M\to N\) and \(g:M\to N\) are smoothly homotopic, then they induce equal maps in de Rham Cohomology.

Raw LaTeX

Let \(M\) and \(N\) be a smooth orientable manifolds. Show that the maps
\(i_a:M\to\R\times M\) given by \(i_a(x)=(a,x)\) and
\(\pi:\R\times M\to M\) given by \(\pi(t,x)=x\) induce inverse maps in de Rham
Cohomology. Conclude that if smooth maps \(f:M\to N\) and \(g:M\to N\) are smoothly
homotopic, then they induce equal maps in de Rham Cohomology.

Topic Tags

Cohomology and Cup Products Homotopy and Degree on Manifolds Orientability and Volume Forms de Rham Cohomology