Realizing doubles: a conjugation zoo. (English) Zbl 1464.55017
Many spaces with a \(C_2\)-action have the feature that their fixed points have the same cohomology up to scaling. For example, complex projective spaces with the complex conjugation action have this feature. Hausmann-Holm-Puppe made this observation precise in [J.-C. Hausmann et al., Algebr. Geom. Topol. 5, 923–964 (2005; Zbl 1081.55006)] and called these spaces ‘conjugation spaces’. In this paper, the authors address two main questions regarding conjugation spaces: the existence of exotic conjugation spaces, and realizability.
A conjugation space is said to be exotic if it is not homotopy equivalent to a conjugation space built from so-called conjugation cells, and they exhibit various examples of these. With regards to the second question, the authors investigate what spaces can be realized as the fixed points of a conjugation space. Their main result in this direction shows that the homotopy type of any simply connected 4 dimensional Poincaré complex can be realized in this way. The results of the paper are illustrated with many interesting examples.
A conjugation space is said to be exotic if it is not homotopy equivalent to a conjugation space built from so-called conjugation cells, and they exhibit various examples of these. With regards to the second question, the authors investigate what spaces can be realized as the fixed points of a conjugation space. Their main result in this direction shows that the homotopy type of any simply connected 4 dimensional Poincaré complex can be realized in this way. The results of the paper are illustrated with many interesting examples.
Reviewer: Jordan Williamson (Praha)
MSC:
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 57S17 | Finite transformation groups |
| 55S10 | Steenrod algebra |
| 55S35 | Obstruction theory in algebraic topology |
Citations:
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