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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 5

If \(X\) and \(Y\) are compact connected smooth \(n\)-manifolds then for \(0<k<n\), the singular homology satisfies \[ H_k(X\times Y,\mathbb {Z})=H_k(X,\mathbb {Z})\oplus H_k(Y,\mathbb {Z}). \]

Raw LaTeX

If \(X\) and \(Y\) are compact connected smooth \(n\)-manifolds then
for \(0<k<n\), the singular homology satisfies
\[
  H_k(X\times Y,\mathbb{Z})=H_k(X,\mathbb{Z})\oplus H_k(Y,\mathbb{Z}).
\]

Topic Tags

Fundamental Groups Homology with Coefficients Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds