Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2014 • Question 9

Question

Math 6510 • Differentiable Manifolds Spring 2014 • Question 9

Let \(Y\) be a space obtained from a space \(X\) by attaching an \(n\)-cell. What is the relationship between the homology groups of \(X\) and of \(Y\)? Specifically, prove the following, where \(f:X\to Y\) is inclusion.

\(f_*:H_i(X)\to H_i(Y)\) is an isomorphism for \(i\ne n-1,n\).
\(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) is surjective.
\(f_*:H_n(X)\to H_n(Y)\) is injective.
Either \(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) has finite kernel or \(f_*:H_n(X)\to H_n(Y)\) is an isomorphism (but not both).

Assuming the ranks of \(H_{n-1}(X)\) and \(H_n(X)\) are finite, also prove that \[ \operatorname {rank}(H_n(X))-\operatorname {rank}(H_{n-1}(X)) =\operatorname {rank}(H_n(Y))-\operatorname {rank}(H_{n-1}(Y))-1. \]

Let \(Y\) be a space obtained from a space \(X\) by attaching an \(n\)-cell. What
is the relationship between the homology groups of \(X\) and of \(Y\)? Specifically,
prove the following, where \(f:X\to Y\) is inclusion.
\begin{itemize}
\item \(f_*:H_i(X)\to H_i(Y)\) is an isomorphism for \(i\ne n-1,n\).
\item \(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) is surjective.
\item \(f_*:H_n(X)\to H_n(Y)\) is injective.
\item Either \(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) has finite kernel or
\(f_*:H_n(X)\to H_n(Y)\) is an isomorphism (but not both).
\end{itemize}
Assuming the ranks of \(H_{n-1}(X)\) and \(H_n(X)\) are finite, also prove that
\[
  \rank(H_n(X))-\rank(H_{n-1}(X))
  =\rank(H_n(Y))-\rank(H_{n-1}(Y))-1.
\]

Topic Tags

CW Complexes and Cell Attachments Cellular Homology Homology with Coefficients Relative Homology and Long Exact Sequences