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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 12

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 12

Suppose \(X\) is a connected 2-dimensional CW complex such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\). Compute \(H_i(X;\mathbb {Z}_2)\), \(H^i(X;\mathbb {Z})\) and \(H^i(X;\mathbb {Z}_2)\) for all \(i\in \mathbb {Z}\).

Raw LaTeX

Suppose \(X\) is a connected 2-dimensional CW complex such that
\(H_1(X;\Z)\cong\Z/2\) and \(H_2(X;\Z)\cong\Z\). Compute
\(H_i(X;\Z_2)\), \(H^i(X;\Z)\) and \(H^i(X;\Z_2)\) for all \(i\in\Z\).

Topic Tags

CW Complexes and Cell Attachments Cohomology and Cup Products Fundamental Groups Homology with Coefficients