Timeline for When are the homology and cohomology Hopf algebras of topological groups equal?
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4 events
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| Jun 28, 2010 at 2:29 | comment | added | Tyler Lawson | The H-space BO, using the multiplication induced by Whitney sum, has homology and cohomology both equal to polynomial algebras - the Stiefel-Whitney classes are not primitive as one can see from the Cartan formula. The same is true for BU integrally. These are not topological groups, but there are topological groups which are homotopy equivalent to these as H-spaces. | |
| Jun 17, 2010 at 16:52 | comment | added | Allen Hatcher | That's right, I should have mentioned the condition of being primitively generated. I also don't know of topological examples of more complicated sorts of duality relationships (but I'm not an expert in this area). | |
| Jun 17, 2010 at 16:42 | comment | added | Torsten Ekedahl | Just to avoid people getting the wrong picture, the dual of a primitively generated polynomial algebra is a divided power algebra. At least algebraically there are Hopf algebra structures on polynomial rings whose duals are not divided power algebras (I don't know if they can be realised as the cohomology of a topological group). | |
| Jun 17, 2010 at 14:16 | history | answered | Allen Hatcher | CC BY-SA 2.5 |