Due Thursday, April 25, in class.

A. Prove the splitting lemma.

p. 176, Hatcher

# 1, 2, 8

Extra hint for 2: it is easy to get confused about the subscripts. Recall that a connected graph has (reduced) nonzero homology only in dimension one. This observation will save you a lot of trouble when considering Mayer-Vietoris sequences. Once you have figured out the case where \(X\) is a tree, build up to arbitrary graphs by adding edges one at a time.

Due in class April 18.

p 155, section 2.2: 21, 22, 28a, 29

A. Suppose we have a smooth embedding \(f: S^1\times D^2 \hookrightarrow S^3\). Let \(K = f(S^1\times 0)\) and \(N(K) = f(S^1\times D^2)\). We call \(K\) a knot and \(N(K)\) its tubular neighborhood. We call \(S^3 – K\) the knot complement. Compute \(H_n(S^3 – K)\) for all \(n\) using Mayer-Vietoris. Carefully specify generators of all spaces involved.

B. Consider \(\mathbb{R}^3 – K \subset S^3 – K \). Using a Mayer-Vietoris sequence, find a generator for \(H_2(\mathbb{R}^3-K)\).

C. Write a short essay outlining a proof that for a homology theory \(h\) on the category of finite CW pairs, the group \(h_0(\ast)\) determines all homology groups \(h_n(X,A)\) for all CW pairs \((X,A)\). A flow chart may be useful.

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.