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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2017 • Question 8

Question

Math 6510 • Differentiable Manifolds Summer 2017 • Question 8

For \(k\in \N \) and a topological space \(X\), let \(C_kX\) be a collection of \(k\) cones on \(X\), with their bases – each homeomorphic to \(X\) – identified. So for example, \(C_2X\) is the suspension of \(X\). Express the homology groups (with integer coefficients) of \(C_kX\) in terms of the homology groups of \(X\).

Raw LaTeX

For \(k\in\N\) and a topological space \(X\), let \(C_kX\) be a collection of
\(k\) cones on \(X\), with their bases -- each homeomorphic to \(X\) -- identified. So
for example, \(C_2X\) is the suspension of \(X\). Express the homology groups (with
integer coefficients) of \(C_kX\) in terms of the homology groups of \(X\).

Topic Tags

Cellular Homology Homology with Coefficients Point-Set Topology and Separation Theorems Smooth Manifolds and Embedded Submanifolds