This assignment is due April 11 in class.

A. Let \(M_2 \) denote the orientable genus two surface, and let \(X\) denote \(M_2 – \mathrm{int\ } D^2\). Call the boundary curve \(\gamma\). Form a space \(Y\) by attaching a disk \(D^2\) along its boundary to the curve \(\gamma\) by a map of degree \(5\). Compute the homology groups of \(Y\).

B. Let \(X = S^1 \vee S^1\), the circles labeled \(a\) and \(b\), and form \(Y\) by gluing two 2-cells onto \(X\) by the identifications \(a^4\) and \(a^4b^{-2}a^2b^2a^{-2}\). Compute the homology groups of \(Y\).

C. Recall that \(\mathbb{R}P^3 = e^0\cup e^1 \cup e^2 \cup e^3\). Let \(X\) be the space obtained by attaching a 4-cell to \(\mathbb{R}P^3\) where the composition of the quotient map and attaching map \(\Delta: S^3 \xrightarrow{\phi} \mathbb{R}P^3 \xrightarrow{q} \overline{e^3}/\partial{e^3} \cong S^3\) has degree \(3\). Compute the homology groups of \(X\).

D. Recall that \(\mathbb{R}P^3\) is naturally a subcomplex of \(\mathbb{R}P^4\). Compute the homology of \(\mathbb{R}P^4/ \mathbb{R}P^3\) using cellular homology.