Universal Coefficient Theorem

1. Construct a single chain complex \(C\) such that \(C\) has the following three properties:

   • \(H_i(C)\) is non-zero if and only \(i=1\).
   • There is a coefficient group \(G_0\) such that \(\tilde {H}^i(C;G_0)=0\) for all \(i\).
   • There is a coefficient group \(G_1\) such that \(\tilde {H}^i(C;G_1)\) is non- zero if and only if \(i=1,2\).
2. Find a topological space \(X\) such that \(H_i(C)=\tilde {H}_i(X)\) for all \(i\).

Show that the union of circles in \(\mathbb {R}^2\) with centers at \((0,\frac {1}{2n})\) and radius \(\frac {1}{n}\) does not have a universal cover.

What is the universal cover of \(\mathbb {R}\mathrm {P}^2\vee \mathbb {R}\mathrm {P}^2\)?

What is the universal cover of \(S^1\vee S^1\)?

Let \(X\) be a path connected space that is locally simply connected. Give the construction of \(p:\widetilde {X}\to X\), the universal cover of \(X\). You don’t have to define what a universal cover is, or show that your construction is a universal cover. Just state the description of the universal cover for \(X\) used in the standard proof that the universal cover for \(X\) exists.

Let \(G\) be a Lie group, and let \(\widetilde {G}\) be the universal cover of \(G\). Endow \(\widetilde {G}\) with the structure of a Lie group.

1. State the universal coefficient theorem for cohomology.
2. Suppose \(X\) is a space such that \(H_0(X)=\mathbb {Z}\), \(H_1(X)=\mathbb {Z}/2\mathbb {Z}\) and \(H_i(X)=0\) for \(i>1\). Compute \(H^i(X;\mathbb {Z}/2\mathbb {Z})\) for all \(i\).

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.

Give an example of a topological space that does not have a universal cover.