Mayer-Vietoris Sequences

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 \(X\) denote the solid torus \(S^1\times D^2\). Using Mayer-Vietoris, compute the homology groups of the double of \(X\), obtained by taking two copies of \(X\) and gluing them along the boundary via the identity map. Also compute \(H_i(X,\partial X)\).

Let \(X\) be the space obtained from two \(n\)-spheres by identifying them along their equatorial \((n-1)\)-spheres. Using any method you like compute \(H_i(X)\) for all \(i\). State any theorems you are using (e.g. Mayer-Vietoris).

Use the Mayer-Vietoris sequence to compute homology groups (with \(\mathbb {Z}\)-coefficients) of the space \(X_{2,3}\) from problem #17. Do the same for the space \(X_{2,2}\) defined similarly except that on \(S^1\times \{1\}\) antipodal points are identified.

Compute de Rham cohomology of the 2-torus. You may use the Poincare lemma and the Mayer-Vietoris sequence. For example, you may want to compute \(H^*(S^1)\) first.

The surface \(M_g\) of genus \(g\), embedded in \(\mathbb {R}^3\) in the standard way, bounds a compact region \(R\). Two copies of \(R\), glued together by the identity map between their boundary surfaces \(M_g\), form a closed 3-manifold \(X\). Compute the homology groups of \(X\) via the Mayer-Vietoris sequence for this decomposition of \(X\) into two copies of \(R\).

Diagram: a standard embedded genus \(g\) surface, drawn schematically as a handled surface bounding a compact region \(R\).

Suppose that \(K_1\supset K_2\supset K_3\supset \cdots \) is a nested sequence of compact connected subsets of \(S^2\) such that \(\pi _1(K_j)=\mathbb {Z}\) for all \(j\). Prove or provide a counterexample for each of the following statements:

1. \(\bigcap _{j=1}^{\infty }K_j\) is connected.
2. \(\pi _1\left (\bigcap _{j=1}^{\infty }K_j\right )\cong \mathbb {Z}\).

Consider the space \(\mathcal {L}\) of all straight lines \(ax+by+c=0\) in the Euclidean plane \(\mathbb {R}^2\). Topologize \(\mathcal {L}\) so that a sequence of lines \(L_i\) is convergent to a line \(L=\{ax+by+c=0\}\) if and only if \(L_i\) can be represented by equations \(a_ix+b_iy+c_i=0\) so that \(\lim _i a_i=a\), \(\lim _i b_i=b\), and \(\lim _i c_i=c\). Prove that the topological space \(\mathcal {L}\) is homeomorphic to the Moebius band without boundary.