Geometric cohomology frames on Hausmann–Holm–Puppe conjugation spaces
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- by Joost van Hamel PDF
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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.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1557-1564
- MSC (2000): Primary 55M35, 55N91, 57S17, 57R91
- DOI: https://doi.org/10.1090/S0002-9939-06-08638-2
- MathSciNet review: 2276667