Math 6510 • Differentiable Manifolds

Below is a picture of a \(\Delta\)-complex \(X\) obtained from the square
subdivided with the diagonal by identifying all sides of the square with
orientations as indicated. So \(X\) has one vertex, two edges and two 2-cells.
Write down the simplicial cochain complex that computes \(H^i(X;\Z/2\Z)\).
Compute the algebra \(H^*(X;\Z/2\Z)\) using the definition of the cup product.
Note: \(X\) is homotopy equivalent to a familiar space but you are not allowed to
go via that route.

\begin{center}
  \includegraphics[
    alt={A square subdivided by a diagonal from the lower-left corner to the upper-right corner. The top and right edges are oriented to the right/up under their side identifications, the bottom and left edges are oriented left/up, and the diagonal is oriented from lower-left to upper-right.},
    width=0.22\textwidth
  ]{GeomTopSu2020-diagram-12.png}
\end{center}