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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2014 • Question 11

Question

Math 6510 • Differentiable Manifolds Fall 2014 • Question 11

Suppose a space \(X\) can be written as the union \(\bigcup _{i=1}^n U_i\) of open subsets \(U_i\). Also assume that every \(U_i\), as well as every nonempty intersection \(U_{i_1}\cap U_{i_2}\cap \cdots \cap U_{i_k}\), is contractible. Prove that \(\widetilde {H}_i(X;\mathbb {Z})=0\) for \(i\ge n-1\). (Hint: For \(i=1,\ldots ,n-1\) let \(V_i=U_n\cap U_i\) and apply induction.)

Raw LaTeX

Suppose a space \(X\) can be written as the union \(\bigcup_{i=1}^n U_i\) of open
subsets \(U_i\). Also assume that every \(U_i\), as well as every nonempty intersection
\(U_{i_1}\cap U_{i_2}\cap\cdots\cap U_{i_k}\), is contractible. Prove that
\(\widetilde{H}_i(X;\Z)=0\) for \(i\ge n-1\). (Hint: For \(i=1,\ldots,n-1\) let
\(V_i=U_n\cap U_i\) and apply induction.)

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Transversality and Intersection Theory