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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2012 • Question 12

Question

Math 6510 • Differentiable Manifolds Spring 2012 • Question 12

Let \(X\) be a topological space and \(x\in X\). Assume that for some topological space \(L\) the point \(x\) has a neighborhood \(U\) in \(X\) which is homeomorphic to the open cone \(cL=L\times [0,\infty )/L\times \{0\}\) by a homeomorphism that takes \(x\) to \(L\times \{0\}\). Prove that \(H_k(X,X-\{x\})\cong \widetilde {H}_{k-1}(L)\).

Raw LaTeX

Let \(X\) be a topological space and \(x\in X\). Assume that for some
topological space \(L\) the point \(x\) has a neighborhood \(U\) in \(X\) which is
homeomorphic to the open cone \(cL=L\times[0,\infty)/L\times\{0\}\) by a
homeomorphism that takes \(x\) to \(L\times\{0\}\). Prove that
\(H_k(X,X-\{x\})\cong\widetilde{H}_{k-1}(L)\).

Topic Tags

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