Show that if \(X\) and \(Y\) are finite CW complexes such that \(H^*(X; \mathbb{Z})\) and \(H^*(Y; \mathbb{Z})\) contain no elements of order a power of a given prime \(p\), then the same is true for \(X \times Y\). [Apply Theorem 3.15 with coefficients in various fields.]
Solution
1. For a finite CW complex \(Z\), \(H^*(Z;\ZZ)\) has no \(p\)-torsion iff \(\dim_{\FF_p} H^n(Z;\FF_p) = \dim_{\QQ} H^n(Z;\QQ)\) for all \(n\).
1.1. By the universal coefficient theorem, \(H^n(Z;\FF_p) \cong H^n(Z;\ZZ) \otimes \FF_p \oplus \operatorname{Tor}(H^{n+1}(Z;\ZZ), \FF_p)\).
UCT for cohomology.
1.2. \(\operatorname{Tor}(H^{n+1}(Z;\ZZ), \FF_p)\) is the \(p\)-torsion of \(H^{n+1}(Z;\ZZ)\).
\(\operatorname{Tor}(A, \FF_p) \cong \{a \in A : pa = 0\}\).
1.3. \(\dim_{\FF_p}(H^n(Z;\ZZ) \otimes \FF_p) = \operatorname{rank} H^n(Z;\ZZ) = \dim_{\QQ} H^n(Z;\QQ)\).
tensoring a finitely generated abelian group with \(\FF_p\) kills the torsion and keeps the free part; the rank equals the rational Betti number.
1.4. Hence \(\dim_{\FF_p} H^n(Z;\FF_p) = \dim_{\QQ} H^n(Z;\QQ) + \dim_{\FF_p}(p\text{-torsion of } H^{n+1}(Z;\ZZ))\).
<2>1–<2>3.
1.5. Therefore the equality \(\dim_{\FF_p} H^n(Z;\FF_p) = \dim_{\QQ} H^n(Z;\QQ)\) for all \(n\) holds iff \(H^*(Z;\ZZ)\) has no \(p\)-torsion.
4.
2. By hypothesis, \(\dim_{\FF_p} H^i(X;\FF_p) = \dim_{\QQ} H^i(X;\QQ)\) and \(\dim_{\FF_p} H^j(Y;\FF_p) = \dim_{\QQ} H^j(Y;\QQ)\) for all \(i, j\).
1 applied to \(X\) and \(Y\).
3. \(\dim_{\FF_p} H^n(X \times Y;\FF_p) = \sum_{i+j=n} \dim_{\FF_p} H^i(X;\FF_p) \cdot \dim_{\FF_p} H^j(Y;\FF_p)\).
Künneth theorem (Theorem 3.15) with field coefficients \(\FF_p\); the Tor term vanishes over a field.
4. \(\dim_{\QQ} H^n(X \times Y;\QQ) = \sum_{i+j=n} \dim_{\QQ} H^i(X;\QQ) \cdot \dim_{\QQ} H^j(Y;\QQ)\).
Künneth theorem with field coefficients \(\QQ\).
5. Hence \(\dim_{\FF_p} H^n(X \times Y;\FF_p) = \dim_{\QQ} H^n(X \times Y;\QQ)\) for all \(n\).
3 and 4 have equal summands term-by-term by 2.
6. Therefore \(H^*(X \times Y;\ZZ)\) has no \(p\)-torsion.
5 and 1 applied to \(Z = X \times Y\).
7. Q.E.D.
6.