Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2020 • Question 9

Question

Math 6510 • Differentiable Manifolds Spring 2020 • Question 9

Recall that if \(X\) is a topological space its suspension \(SX\) is the product \(X\times [0,1]\) with both \(X\times \{0\}\) and \(X\times \{1\}\) collapsed to points. Show that \(\operatorname {H}_{n+1}(SX)=\operatorname {H}_n(X)\) for \(n\ge 0\).

Raw LaTeX

Recall that if \(X\) is a topological space its suspension \(SX\) is the
product \(X\times[0,1]\) with both \(X\times\{0\}\) and \(X\times\{1\}\) collapsed to
points. Show that \(\HH_{n+1}(SX)=\HH_n(X)\) for \(n\ge0\).

Topic Tags

Cohomology and Cup Products Homology with Coefficients Point-Set Topology and Separation Theorems Pullbacks and Wedge Products