Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:11:35Z http://mathoverflow.net/feeds/question/57120 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57120/spaces-which-have-the-same-homology-groups-the-same-cohomology-groups-but-have Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? rhl 2011-03-02T15:56:28Z 2011-03-02T18:00:40Z <p>Does anyone have an example of two spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?</p> <p>is it possible?</p> http://mathoverflow.net/questions/57120/spaces-which-have-the-same-homology-groups-the-same-cohomology-groups-but-have/57121#57121 Answer by Neil Strickland for Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? Neil Strickland 2011-03-02T15:59:17Z 2011-03-02T15:59:17Z <p>A very standard example would be $S^2\vee S^4$ and $\mathbf{C}P^2$.</p> http://mathoverflow.net/questions/57120/spaces-which-have-the-same-homology-groups-the-same-cohomology-groups-but-have/57144#57144 Answer by damiano for Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? damiano 2011-03-02T18:00:40Z 2011-03-02T18:00:40Z <p>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.</p>