Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings? - MathOverflow most recent 30 from http://mathoverflow.net2013-05-24T18:11:35Zhttp://mathoverflow.net/feeds/question/57120http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/57120/spaces-which-have-the-same-homology-groups-the-same-cohomology-groups-but-haveSpaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?rhl2011-03-02T15:56:28Z2011-03-02T18:00:40Z
<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#57121Answer by Neil Strickland for Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?Neil Strickland2011-03-02T15:59:17Z2011-03-02T15:59:17Z<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#57144Answer by damiano for Spaces which have the same homology groups, the same cohomology groups, but have different cohomology rings?damiano2011-03-02T18:00:40Z2011-03-02T18:00:40Z<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>