Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6520 • Algebraic Topology

Summer 2021 • Question 6

Question

Math 6520 • Algebraic Topology Summer 2021 • Question 6

  1. Define the notion of a chain morphism between two chain complexes and prove that it induces a homomorphism in homology.
  2. Define the notion of a chain homotopy equivalence.
  3. Prove that a chain homotopy equivalence induces an isomorphism in homology.

Raw LaTeX

\begin{enumerate}

\item[(i)]
Define the notion of a chain morphism between two chain complexes
and prove that it induces a homomorphism in homology.

\item[(ii)]
Define the notion of a chain homotopy equivalence.
\item[(iii)]
Prove that a chain homotopy equivalence induces an isomorphism
in homology.

\end{enumerate}

Topic Tags

CW Complexes and Cell Attachments Chain Complexes and Chain Homotopy Homology with Coefficients Homotopy Equivalence, Retracts, and Contractibility