Compute \(H_*(\Sigma\RP^2 \cross \RP^2; \ZZ)\)
Solution
Goal: Compute the integral homology groups \(H_k(\Sigma \mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z})\) for all \(k \ge 0\).
1. Compute the homology of the factors \(\Sigma \mathbb{RP}^2\) and \(\mathbb{RP}^2\).
1.1. The homology of the real projective plane \(\mathbb{RP}^2\) is:
\(H_0(\mathbb{RP}^2) \cong \mathbb{Z}\),
\(H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}\),
\(H_2(\mathbb{RP}^2) = 0\),
\(H_k(\mathbb{RP}^2) = 0\) for \(k \ge 3\). <2>2. By the suspension isomorphism \(\widetilde{H}_k(\Sigma X) \cong \widetilde{H}_{k-1}(X)\), the reduced homology of \(\Sigma \mathbb{RP}^2\) is:
\(H_0(\Sigma \mathbb{RP}^2) \cong \mathbb{Z}\),
\(H_1(\Sigma \mathbb{RP}^2) \cong \widetilde{H}_0(\mathbb{RP}^2) = 0\),
\(H_2(\Sigma \mathbb{RP}^2) \cong \widetilde{H}_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}\),
\(H_3(\Sigma \mathbb{RP}^2) \cong \widetilde{H}_2(\mathbb{RP}^2) = 0\),
\(H_k(\Sigma \mathbb{RP}^2) = 0\) for \(k \ge 4\). <2>3. Proof: Standard cellular homology of \(\mathbb{RP}^2\) and suspension theorem. Q.E.D.
2. Apply the Künneth formula for homology.
2.1. For topological spaces \(A\) and \(B\), the Künneth formula over PID \(\mathbb{Z}\) gives a split short exact sequence: \[0 \to \bigoplus_{i+j=k} (H_i(A) \otimes_\mathbb{Z} H_j(B)) \to H_k(A \times B) \to \bigoplus_{i+j=k-1} \operatorname{Tor}_1^\mathbb{Z}(H_i(A), H_j(B)) \to 0.\]
2.2. Here \(A = \Sigma \mathbb{RP}^2\) and \(B = \mathbb{RP}^2\).
2.3. Tensor products \(H_i(A) \otimes H_j(B)\):
\(k = 0\): \(H_0(A) \otimes H_0(B) = \mathbb{Z} \otimes \mathbb{Z} \cong \mathbb{Z}\).
\(k = 1\): \((H_0(A) \otimes H_1(B)) \oplus (H_1(A) \otimes H_0(B)) = (\mathbb{Z} \otimes \mathbb{Z}/2) \oplus (0 \otimes \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}\).
\(k = 2\): \((H_0(A) \otimes H_2(B)) \oplus (H_1(A) \otimes H_1(B)) \oplus (H_2(A) \otimes H_0(B)) = 0 \oplus 0 \oplus (\mathbb{Z}/2 \otimes \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}\).
\(k = 3\): \((H_2(A) \otimes H_1(B)) \oplus (H_3(A) \otimes H_0(B)) = (\mathbb{Z}/2 \otimes \mathbb{Z}/2) \oplus 0 \cong \mathbb{Z}/2\mathbb{Z}\).
\(k \ge 4\): All tensor terms are 0. <2>4. Tor terms \(\operatorname{Tor}_1^\mathbb{Z}(H_i(A), H_j(B))\):
Recall \(\operatorname{Tor}(\mathbb{Z}, -) = 0\) and \(\operatorname{Tor}(\mathbb{Z}/2, \mathbb{Z}/2) \cong \mathbb{Z}/2\mathbb{Z}\).
For \(i+j = 3\), the only non-zero term is \(i = 2, j = 1\): \(\operatorname{Tor}_1(H_2(A), H_1(B)) = \operatorname{Tor}_1(\mathbb{Z}/2, \mathbb{Z}/2) \cong \mathbb{Z}/2\mathbb{Z}\). This contributes to \(k = (i+j) + 1 = 4\).
For all other \(i, j\), at least one factor is free (\(\mathbb{Z}\) or \(0\)), so all other Tor terms vanish.
2.5. The Künneth formula expresses \(H_k(A \times B)\) as the direct sum of the tensor terms over \(i + j = k\) and the Tor terms over \(i + j = k - 1\); the computations in <2>3–<2>4 evaluate each of these terms.
3. Combine terms for each dimension \(k\).
3.1. \(k = 0\): \(H_0 \cong \mathbb{Z}\).
3.2. \(k = 1\): \(H_1 \cong \mathbb{Z}/2\mathbb{Z}\).
3.3. \(k = 2\): \(H_2 \cong \mathbb{Z}/2\mathbb{Z}\).
3.4. \(k = 3\): \(H_3 \cong \mathbb{Z}/2\mathbb{Z}\).
3.5. \(k = 4\): \(H_4 \cong \operatorname{Tor}_1(H_2(A), H_1(B)) \cong \mathbb{Z}/2\mathbb{Z}\).
3.6. \(k \ge 5\): \(H_k = 0\).
3.7. Each \(H_k\) is the direct sum of the tensor terms (from <1>2.<2>3) and the Tor terms (from <1>2.<2>4) in that degree; summing them gives the groups listed in <2>1–<2>6.
4. Conclusion.
4.1. The homology groups are: \[H_k(\Sigma \mathbb{RP}^2 \times \mathbb{RP}^2; \mathbb{Z}) \cong \begin{cases} \mathbb{Z} & k = 0, \\ \mathbb{Z}/2\mathbb{Z} & k = 1, 2, 3, 4, \\ 0 & k \ge 5. \end{cases}\]
4.2. The groups computed in 3 assemble into the stated description of \(H_k\) for all \(k \ge 0\).