6520-2023-08

Let \(a\) and \(b\) be the two generators of \(\pi _1(S^1 \vee S^1)\) corresponding to the two \(S^1\) summands. Draw a picture of the covering space of \(S^1 \vee S^1\) corresponding to the normal subgroup generated by \(a^2\), \(b^2\), and \((ab)^4\), and prove this covering space is indeed the correct one.

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T = S^1 \times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1 \times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).

2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) coefficients.

Let \(\Sigma _g\) be the closed, orientable, genus-\(g\) surface. Show that \(\pi _1(\Sigma _2)\) contains \(\pi _1(\Sigma _4)\) as a normal subgroup.

Prove or disprove:

1. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(f_d : T^2 \to S^2\).

2. For every \(d \in \mathbb {Z}\) there is a degree \(d\) map \(g_d : S^2 \to T^2\).

   Here \(S^2\) is the 2-sphere and \(T^2 = S^1 \times S^1\) is the 2-dimensional torus.

Let \(M\) be a closed, orientable 5-manifold with \(H_1(M) = \mathbb {Z} \oplus \mathbb {Z}_5\) and \(H_2(M) = \mathbb {Z}_2\). Find \(H_i(M)\) and \(H^i(M;\mathbb {Z})\) for all \(i \ge 0\).

Construct a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to calculate the cohomology groups \(H^i(\mathbb {R}P^2;\mathbb {Z}_2)\) and the cup product structure.