Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2002 • Question 20

Question

Math 6510 • Differentiable Manifolds Fall 2002 • Question 20

Let \(X\) be a topological space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=0\), \(H_2(X)=\mathbb {Z}_2\), \(H_3(X)=\mathbb {Z}_3\), \(H_4(X)=\mathbb {Z}\), \(H_k(X)=0\), \(k>4\). Compute \(H_*(X;\mathbb {Z}_2)\) and \(H^*(X;\mathbb {Z})\).

Raw LaTeX

Let \(X\) be a topological space such that \(H_0(X)=\mathbb{Z}\), \(H_1(X)=0\),
\(H_2(X)=\mathbb{Z}_2\), \(H_3(X)=\mathbb{Z}_3\), \(H_4(X)=\mathbb{Z}\),
\(H_k(X)=0\), \(k>4\). Compute \(H_*(X;\mathbb{Z}_2)\) and \(H^*(X;\mathbb{Z})\).

Topic Tags

Cohomology and Cup Products Fundamental Groups Homology with Coefficients Point-Set Topology and Separation Theorems