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Realizing doubles: a conjugation zoo

Part of: Homotopy theory Operations and obstructions

Published online by Cambridge University Press:  08 April 2020

Wolfgang Pitsch  and
Jérôme Scherer
Wolfgang Pitsch
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193Bellaterra, Spain (pitsch@mat.uab.es)
Jérôme Scherer
Affiliation:
EPFL, Mathematics, Station 8, CH-1015Lausanne, Switzerland (jerome.scherer@epfl.ch)
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Abstract

Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod 2 cohomology (as a graded vector space, a ring and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct ‘exotic’ conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.

Keywords

Conjugation spacesrealizationHopf invariant

MSC classification

Primary: 55P91: Equivariant homotopy theory
Secondary: 57S17: Finite transformation groups 55S10: Steenrod algebra 55S35: Obstruction theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 509 - 524
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

1Adams, J. F.. On the non-existence of elements of Hopf invariant one. Ann. Math. 72 (1960), 20–104.CrossRefGoogle Scholar
2Araki, S. and Iriye, K.. Equivariant stable homotopy groups of spheres with involutions. I. Osaka Math. J. 19 (1982), 1–55.Google Scholar
3Berrick, A. J. and Casacuberta, C.. A universal space for plus-constructions. Topology 38 (1999), 467–477.CrossRefGoogle Scholar
4Bousfield, A. K. and Kan, D. M.. Homotopy limits, completions and localizations. Lecture Notes in Mathematics, vol. 304 (Berlin: Springer-Verlag, 1972).CrossRefGoogle Scholar
5Cannon, J. W.. Shrinking cell-like decompositions of manifolds. Codimension three. Ann. Math. 110 (1979), 83–112.CrossRefGoogle Scholar
6Dickson, L. E.. On quaternions and their generalization and the history of the eight square theorem. Ann. Math. 20 (1919), 155–171.CrossRefGoogle Scholar
7Dold, A.. Erzeugende der thomschen algebra ${\mathfrak {N}}$. Math. Z. 65 (1956), 25–35.CrossRefGoogle Scholar
8Dugger, D. and Isaksen, D. C.. $\mathbb {Z}/2$-equivariant and $\mathbb {R}$-motivic stable stems. Proc. Am. Math. Soc. 145 (2017), 3617–3627.CrossRefGoogle Scholar
9Floyd, E. E.. The number of cells in a non-bounding manifold. Ann. Math. 98 (1973), 210–225.CrossRefGoogle Scholar
10Franz, M. and Puppe, V.. Steenrod squares on conjugation spaces. C. R. Math. Acad. Sci. Paris 342 (2006), 187–190.CrossRefGoogle Scholar
11Hambleton, I. and Hausmann, J.-C.. Conjugation spaces and 4-manifolds. Math. Z. 269 (2011), 521–541.CrossRefGoogle Scholar
12Hausmann, J.-C., Holm, T. and Puppe, V.. Conjugation spaces. Algebr. Geom. Topol. 5 (2005), 923–964.CrossRefGoogle Scholar
13Hill, M. A., Hopkins, M. J. and Ravenel, D. C.. On the nonexistence of elements of Kervaire invariant one. Ann. Math. 184 (2016), 1–262.CrossRefGoogle Scholar
14Hilton, P. J.. On the homotopy groups of the union of spheres. J. London Math. Soc. 30 (1955), 154–172.CrossRefGoogle Scholar
15Hurwitz, A.. Über die komposition der quadratischen formen. Math. Ann. 88 (1922), 1–25.CrossRefGoogle Scholar
16Milnor, J. and Husemoller, D.. Symmetric bilinear forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73 (New York-Heidelberg: Springer-Verlag, 1973).CrossRefGoogle Scholar
17Mukai, J. and Shinpo, T.. Some homotopy groups of the mod 4 Moore space. J. Fac. Sci. Shinshu Univ. 34 (1999), 1–14.Google Scholar
18Olbermann, M.. Conjugations on 6-manifolds. Math. Ann. 342 (2008), 255–271.CrossRefGoogle Scholar
19Olbermann, M.. Involutions on S 6 with 3-dimensional fixed point set. Algebr. Geom. Topol. 10 (2010), 1905–1932.CrossRefGoogle Scholar
20Pitsch, W., Ricka, N. and Scherer, J.. Conjugation spaces are cohomologically pure. ArXiv e-prints, available at https://arxiv.org/abs/1908.03088.Google Scholar
21Pitsch, W. and Scherer, J.. Conjugation spaces and equivariant Chern classes. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 77–90.CrossRefGoogle Scholar
22Schwartz, L.. Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture. Chicago Lectures in Mathematics (Chicago, IL: University of Chicago Press, 1994).Google Scholar
23Sieradski, A. J.. Stabilization of self-equivalences of the pseudoprojective spaces. Michigan Math. J. 19 (1972), 109–119.CrossRefGoogle Scholar
24Thom, R.. Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
25Wall, C. T. C.. Surgery on compact manifolds. London Mathematical Society Monographs, No. 1. (London-New York: Academic Press, 1970).Google Scholar
26Whitehead, G. W.. Elements of homotopy theory. Graduate Texts in Mathematics, vol. 61 (New York-Berlin: Springer-Verlag, 1978).Google Scholar
27Wu, J.. Homotopy theory of the suspensions of the projective plane. Mem. Am. Math. Soc. 162 (2003), no. 769, x+130 pp.Google Scholar