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For any $n > 2$, $\mathbb{CP}^n/\mathbb{CP}^{n-2}$ and $S^{2n}\vee S^{2n-2}$ have the same cohomology groups: for any ring $R$, we have

$$H^k(\mathbb{CP}^n/\mathbb{CP}^{n-2}; R) \cong H^k(S^{2n}\vee S^{2n-2}; R) \cong \begin{cases} R & k = 0, 2n-2, 2n\\ 0 & \text{otherwise}. \end{cases}$$

Moreover, the two spaces have the same cohomology ring structure (the product of any two elements of positive degree is necessarily zero).

However, for $n$ even, the two spaces are not homotopy equivalent. As is discussed here and here, this can be shown by demonstrating that $\operatorname{Sq}^2 : H^{2n-2}(\mathbb{CP}^n/\mathbb{CP}^{n-2}; \mathbb{Z}_2) \to H^{2n}(\mathbb{CP}^n/\mathbb{CP}^{n-2}; \mathbb{Z}_2)$ is an isomorphism, while $\operatorname{Sq}^2 : H^{2n-2}(S^{2n}\vee S^{2n-2}; \mathbb{Z}_2) \to H^{2n}(S^{2n}\vee S^{2n-2}; \mathbb{Z}_2)$ is the zero map. In particular, we see that the cohomology of the two spaces are not isomorphic as modules over the Steenrod algbera.

I am looking for a similar example where the spaces are closed manifolds.

Are there examples of closed manifolds with isomorphic cohomology rings, but different cohomology modules over the Steenrod algebra?

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  • $\begingroup$ Do you want the cohomology rings to be isomorphic over every ring? $\endgroup$ – user98602 Dec 31 '16 at 17:07
  • $\begingroup$ @MikeMiller: Preferably, but I'd be interested in seeing other examples as well. $\endgroup$ – Michael Albanese Dec 31 '16 at 18:18
  • $\begingroup$ It seems to me that the construction here of a spin and non-spin 5-manifold with second homology $\Bbb Z_2^2$ should furnish examples. There is no third or fourth homology, so over every ring in which 2 is invertible the cohomology rings are clearly the same. Over $\Bbb Z_2$ I think they automatically need to be isomorphic by the nondegeneracy of the cup product. Then the key point is that $w_2$ is zero in one and nonzero in another, and is equal to $v_2$, so that $\text{Sq}^2$ is zero in only one. $\endgroup$ – user98602 Dec 31 '16 at 18:56
  • $\begingroup$ $S^3\times \Bbb CP^n$ and $S^1\times \Bbb CP^n/S^1\times\{x_0\}$ have isomorphic cohomology rings but can be distinguished by their module structure over the Steenrod algebra. $\endgroup$ – iwriteonbananas Jan 1 '17 at 13:09
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Here is an example of two simply connected, closed manifolds $M_0$ and $M_1$ such that $H^*(M_0; R)$ and $H^*(M_1; R)$ are isomorphic as rings for any commutative ring with identity $R$, but $H^*(M_0; \mathbb{Z}_2)$ and $H^*(M_1; \mathbb{Z}_2)$ are not isomorphic as modules over the Steenrod algebra. Moreover, $M_0$ and $M_1$ have the same homotopy groups.

First note that $\operatorname{Vect}^4(S^2) = [S^2, BSO(4)] = \pi_1(SO(4)) \cong \mathbb{Z}_2$. So, up to isomorphism, there are two real rank four vector bundles over $S^2$: the trivial one $E_0 \cong \varepsilon^4$ and a non-trivial one $E_1$. Every rank four bundle over $S^2$ splits as $F\oplus\varepsilon^2$ where $F$ is a real rank two bundle. As $S^2$ is simply connected, $F$ is orientable and can therefore be viewed as a complex line bundle. Identifying $S^2$ with $\mathbb{CP}^1$, we see that $F \cong \mathcal{O}(k)$ for some integer $k$. If $k = 2m + 1$, then $w_2(\mathcal{O}(2m+1)\oplus\varepsilon^2) \neq 0$ so $\mathcal{O}(2m+1)\oplus\varepsilon^2$ is not the trivial bundle and is therefore isomorphic to $E_1$; in particular, we see that $w_2(E_1) \neq 0$. If $k = 2m$, $w_2(\mathcal{O}(2m)\oplus\varepsilon^2) = 0$ so $\mathcal{O}(2m)\oplus\varepsilon^2$ is not isomorphic to $E_1$ and therefore must be trivial.

Equip $E_0$ and $E_1$ with Riemannian metrics, and let $M_0$ and $M_1$ be their corresponding sphere bundles.

There are fibre bundles $S^3 \to M_i \to S^2$. From the long exact sequence in homotopy associated to a fibration, we have an exact sequence

$$\dots \to \pi_2(S^3) \to \pi_2(M_i) \to \pi_2(S^2) \to \pi_1(S^3) \to \pi_1(M_i) \to \pi_1(S^2) \to \dots$$

As $\pi_1(S^2) = \pi_1(S^3) = \pi_2(S^3) = 0$ and $\pi_2(S^2) \cong \mathbb{Z}$, we see that $M_i$ is simply connected and $\pi_2(M_i) \cong \mathbb{Z}$. By the Hurewicz theorem, $H_2(M_i; \mathbb{Z}) \cong \mathbb{Z}$. By the universal coefficient theorem for cohomology, $H^1(M_i; \mathbb{Z}) = 0$ and $H^2(M_i;\mathbb{Z}) \cong \mathbb{Z}$. Since $M_i$ is simply connected, $M_i$ is orientable, so by Poincaré duality $H_3(M_i; \mathbb{Z}) \cong H^2(M_i; \mathbb{Z}) \cong \mathbb{Z}$ and $H_4(M_i; \mathbb{Z}) \cong H^1(M_i; \mathbb{Z}) = 0$. Now by the universal coefficient theorem again, for any commutative ring with identity $R$, we have

$$H^k(M_i; R) \cong \begin{cases} R & k = 0, 2, 3, 5\\ 0 & \text{otherwise}. \end{cases}$$

Alternatively, we could have used the cohomological Serre spectral sequence to arrive at this result. Now Poincaré duality with $R$ coefficients uniquely determines the ring structure on $H^*(M_i; R)$, up to isomorphism, so $H^*(M_0; R) \cong H^*(M_1; R)$ as graded rings.

Let $\tilde{\pi} : E \to B$ be a smooth vector bundle, then $TE \cong \tilde{\pi}^*TB\oplus\tilde{\pi}^*E$. If $i : S(E) \to E$ is the natural inclusion, then $i^*TE = TS(E)\oplus\nu$ where $\nu$ is the normal line bundle. So

$$TS(E)\oplus\nu = i^*TE = i^*\tilde{\pi}^*(TB\oplus E) = (\tilde{\pi}\circ i)^*(TB\oplus E) = \pi^*(TB\oplus E)$$

where $\pi : S(E) \to B$ is the projection of the sphere bundle.

As $M_i$ is simply connected, $\nu$ is trivial; more generally, the normal bundle of a sphere bundle is always trivial, see this note. So

$$w(TM_i) = w(TS(E_i)) = w(\pi^*(TS^2\oplus E_i)) = \pi^*(w(TS^2\oplus E_i)) = \pi^*(w(TS^2)w(E_i)) = \pi^*w(E_i) = \pi^*(1 + w_2(E_i)) = 1 + \pi^*w_2(E_i).$$

It follows from the Gysin sequence that $\pi^* : H^2(S^2; \mathbb{Z}_2) \to H^2(M_i; \mathbb{Z}_2)$ is an isomorphism. Alternatively, we can view this map as an edge homomorphism in the cohomological Serre spectral sequence (see Theorem $5.9$ of A User's Guide to Spectral Sequences by McCleary), from which it is clear the map is an isomorphism. Therefore, $w(M_0) = 1$ while $w(M_1) = 1 + a$ where $a$ is the unique non-zero element of $H^2(M_2; \mathbb{Z}_2)$. So $M_0$ and $M_1$ are not homotopy equivalent. In particular, $\operatorname{Sq}^2 : H^3(M_i; \mathbb{Z}_2) \to H^5(M_i; \mathbb{Z}_2)$ is zero for $i = 0$ and an isomorphism for $i = 1$; that is, $H^*(M_0; \mathbb{Z}_2)$ and $H^*(M_1; \mathbb{Z}_2)$ are not isomorphic as modules over the Steenrod algebra.

Finally, both $E_0$ and $E_1$ admit a nowhere section, so the associated sphere bundles $M_i \to S^2$ admit a section. This implies that the long exact sequence in homotopy of the fibration splits into short exact sequences

$$0 \to \pi_n(S^3) \to \pi_n(M_i) \to \pi_n(S^2) \to 0.$$

Moreover, for $n \geq 2$, $\pi_n(M_i)$ is abelian, so the existence of a right splitting $\sigma_* : \pi_n(S^2) \to \pi_n(M_i)$ implies that the sequence splits, i.e. $\pi_n(M_i) = \pi_n(S^3)\oplus\pi_n(S^2)$. As for $n = 1$, we saw above that $M_0$ and $M_1$ are simply connected.

It should be noted that as $E_0$ is trivial, $E_0 = S^2\times \mathbb{R}^4$ so $M_0 = S^2\times S^3$. With this description, we could have calculated the cohomology ring, Stiefel-Whitney classes, and homotopy groups of $M_0$ using simpler techniques.

Note, the same computation shows that for $k \geq 4$, the sphere bundles of $\varepsilon^k$ and $\mathcal{O}(1)\oplus\varepsilon^{k-2}$ are simply connected, closed $(k + 1)$-dimensional manifolds with isomorphic cohomology rings (for any commutative ring with identity), isomorphic homotopy groups, but are not homotopy equivalent; the first is spin, and the second isn't (and therefore their $\mathbb{Z}_2$ cohomologies are not isomorphic as modules over the Steenrod algebra).

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  • $\begingroup$ Is it easy to see that the non-trivial bundle admits a section? (My obstruction theory is very weak). An alternative computation of the homotopy groups is that any linear $S^k$ bundle over $S^2$ for $k\geq 2$ is the quotient of $S^k\times S^3$ by a circle action, which acts via the Hopf action on the $S^3$ factor and a linear action on the $S^k$ factor. Then the LES assoiated to $S^1\rightarrow S^k\times S^3\rightarrow (S^k\times S^3)/S^1$ gives the result on homotopy groups. $\endgroup$ – Jason DeVito Dec 12 '18 at 17:44
  • $\begingroup$ @JasonDeVito: The non-trivial $S^3$ bundle over $S^2$ is the sphere bundle of $\mathcal{O}(2k+1)\oplus\varepsilon^2$; the former admits a section because the latter admits a nowhere-zero section. $\endgroup$ – Michael Albanese Dec 12 '18 at 18:59
  • $\begingroup$ Oh, duh. Thanks! $\endgroup$ – Jason DeVito Dec 12 '18 at 19:12

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