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Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?

is it possible?

flag	asked Mar 2 2011 at 15:56 rhl 13●3

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sounds like a homework problem to me. – Agol Mar 2 2011 at 17:59

it isn't. I am teaching myself the material. – rhl Mar 2 2011 at 19:40

Slightly different example I quite like: the two partial flag manifolds for $SO(5)$ have the same cohomology groups (free abelian, in even degrees), the same rational cohomology rings, and different integer cohomology rings. – Allen Knutson Mar 4 2011 at 4:07

Neil Strickland · Accepted Answer · 2011-03-02 15:59:17Z UTC

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A very standard example would be $S^2\vee S^4$ and $\mathbf{C}P^2$.

answered Mar 2 2011 at 15:59

Neil Strickland
13.3k●1●28●66

	hm, OK. but these have different homotopy groups, Looks like \pi_5(S^2 \wedge s^4) = Z_2 x Z_2 but \pi_5(CP^2) = Z correct? It's not clear to me that the cohomology ring is actually a finer invariant than cohomology. When I read, all sources say 'the structure is richer than homology,' but is it useful? maybe mathoverflow is not an appropriate forum for this question? – rhl Mar 2 2011 at 19:44
	math.stackexchange.com might be better for you. Also, this is the first example I learned. It satisfies all of the requirements listed in your question. The ring structure on $S^2 \vee S^4$ is trivial, but not on $\mathbb{C}P^2$. – Sean Tilson Mar 2 2011 at 21:43
	oh, and cohomology is easier to compute than homotopy for finite dimensional cell/CW complexes. – Sean Tilson Mar 2 2011 at 21:44

damiano · Answer 2 · 2011-03-02 18:00:40Z UTC

There are also standard examples in which both spaces are compact manifolds. For instance, if $n \geq 1$ is an integer and $Q_n \subset \mathbb{P}^{2n+2}$ is a non-singular quadric, then $Q_n$ has the same integral homology and cohomology groups as $\mathbb{P}^{2n+1}$, but the cohomology rings are different.

Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?

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