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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2013 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 2013 • Question 12

Give an example of a space \(X\) such that \(H_i(X;\mathbb {Z}/2)=0\) for \(i>0\) but for some \(i>0\) \(H_i(X;\mathbb {Z})\ne 0\). Is it possible to have a space \(X\) with \(H_i(X;\mathbb {Z})=0\) for \(i>0\) but for some \(i>0\) \(H_i(X;\mathbb {Z}/2)\ne 0\)? Prove it or give an example.

Raw LaTeX

Give an example of a space \(X\) such that \(H_i(X;\Z/2)=0\) for \(i>0\) but for
some \(i>0\) \(H_i(X;\Z)\ne0\). Is it possible to have a space \(X\) with
\(H_i(X;\Z)=0\) for \(i>0\) but for some \(i>0\) \(H_i(X;\Z/2)\ne0\)? Prove it or give an
example.

Topic Tags

Homology with Coefficients