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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces


Author: Joost van Hamel
Journal: Proc. Amer. Math. Soc. 135 (2007), 1557-1564
MSC (2000): Primary 55M35, 55N91, 57S17, 57R91
Published electronically: October 26, 2006
MathSciNet review: 2276667
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Abstract: For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold $ X$ with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of $ X$), these topological cycles will give rise to a cohomology frame.


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Additional Information

Joost van Hamel
Affiliation: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3001 Leuven (Heverlee), Belgium
Email: vanhamel@member.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08638-2
PII: S 0002-9939(06)08638-2
Received by editor(s): October 7, 2005
Received by editor(s) in revised form: December 23, 2005
Published electronically: October 26, 2006
Communicated by: Paul Goerss
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.