Seifert-van Kampen Theorem

Carefully state Van Kampen’s theorem. The version for two subsets suffices. Use Van Kampen’s theorem to compute \(\pi _1(T^2 \vee T^2)\) where \(T^2\) is a torus.

Let \(x_0\in \mathbb {R}P^2\) be a basepoint and view the wedge \(\mathbb {R}P^2\vee \mathbb {R}P^2\) as the subspace of the product \(\mathbb {R}P^2\times \mathbb {R}P^2\) where at least one coordinate is \(x_0\).

1. Use the Seifert-van Kampen theorem to compute \(\pi _1(\mathbb {R}P^2\vee \mathbb {R}P^2)\).
2. Prove that \(\mathbb {R}P^2\vee \mathbb {R}P^2\) is not a retract of \(\mathbb {R}P^2\times \mathbb {R}P^2\).

Let \(m,n\) be positive integers and view \(S^1\) as the unit circle in the complex plane. Let \(X\) be the space obtained from the cylinder \(S^1\times [0,1]\) by identifying points \((z,0)\) and \((w,0)\) if \(z^n=w^n\), and identifying points \((z,1)\) and \((w,1)\) if \(z^m=w^m\).

1. Use the Mayer-Vietoris sequence to compute \(H_i(X)\) for all \(i\).
2. Use the Seifert-van Kampen theorem to find a presentation for \(\pi _1(X)\).

Let \(X=S^1\vee S^1\), with basepoint \(*\) at the point where the two circles are glued, and with \(a,b:S^1\to X\) the generators of \(\pi _1(X,*)\) corresponding to the identity maps onto each circle in the wedge sum. Construct a covering space \(p:(E,e)\to (X,*)\) such that \(p_*\pi _1(E,e)\) is the subgroup of \(\pi _1(X,*)\) generated by \(a^2\), \(ab^{-1}\), \(ab\). Justify that your answer works in two ways:

1. using covering space theory; and
2. using the van Kampen theorem.

Carefully state the van Kampen theorem.

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 2 attaching map, and the other with degree 4 attaching map. What is the cardinality of the fundamental group of \(X\)? If you use van Kampen’s theorem, state it.