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Marked homeomorphisms and the realization problem

Published online by Cambridge University Press:  24 October 2008

Peter Greenberg
Peter Greenberg
Affiliation:
Université de Grenoble I, Institut Fourier, Laboratoire de Mathématiques, 38402 St Martin d'Hères Cedex (France)
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The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?

The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 4 , May 1996 , pp. 575 - 596
Copyright
Copyright © Cambridge Philosophical Society 1996

References

REFERENCES

[Ber]Berrick, A. J.. An approach to algebraic K-theory (Pitman, 1982).Google Scholar
[Ca]Cartier, P.. Développements récents sur les groupes de tresses: applications à la topologie et à l'algèbre. Sém. Bourbaki, exp. no. 716 (nov. 1989). Astérisque 189–190 (1990), 17–68.Google Scholar
[Co]Cohen, R. L.. The geometry of Ω2S 3 and braid orientations. Invent. Math. 54 (1979), 53–67.CrossRefGoogle Scholar
[Du]Dupont, J. L.. Characteristic classes for flat bundles and their formulas (preprint).Google Scholar
[Gh–Se]Ghys, E. and Sergiescu, V.. Sur un groupe remarquable de difféomorphismes du cercle. Comment. Math. Helv. 62 (1987), 109–138.CrossRefGoogle Scholar
[G1]Greenberg, P.. Classifying spaces for foliations with isolated singularities. Trans. A.M.S. 304 (1987), 417–429.CrossRefGoogle Scholar
[G2]Greenberg, P.. Generators and relations in the classifying space for pl foliations. Top. and its Appl. 48 (1992), 185–205.CrossRefGoogle Scholar
[G3]Greenberg, P.. Area preserving pl homeomorphisms and relations in K2 (to appear).Google Scholar
[G–Se]Greenberg, P. and Sergiescu, V.. A non-commutative Steinberg extension. Prépublication de l'Institut Fourier no. 230, Grenoble, 1993, to appear in J. of K theory.Google Scholar
[Haef1]Haefliger, A.. Homotopy and integrability. Lecture Notes in Math. (Springer, 1971) 197, 133–163.Google Scholar
[Haef2]Haefliger, A.. Complexes of groups and orbihedra (preprint).Google Scholar
[KS]Kirby, R. and Siebenmann, L.. Foundational essays on topological manifolds. Ann. of Math. Studies 88 (P.U. Press, 1977).Google Scholar
[KL]Kuiper, N. and Lashof, R.. Mierobundles and bundles I. Invent. Math. 1 (1966), 1–17.CrossRefGoogle Scholar
[Kan]Kan, D. M.. On c.s.s. complexes. Amer. J. Math. 79 (1957), 449–476.CrossRefGoogle Scholar
[Law]Lawson, H. B.. The qualitative theory of foliations, Regional Conference Series in Mathematics 27 (A.M.S., 1977).Google Scholar
[Math1]Mather, J.. The vanishing of the homology of certain groups of homeomorphisms. Topology 10 (1971), 297–298.CrossRefGoogle Scholar
[Math2]Mather, J.. Integrability in codimension one. Comment. Math. Helv. 48 (1973), 195–233.CrossRefGoogle Scholar
[May1]May, J. P.. The geometry of iterated loop spaces, Lecture Notes in Math. 271 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[May2]May, J. P.. Simplicial objects in algebraic topology (Van Nostrand, 1967).Google Scholar
[May3]May, J. P.. Classifying spaces and fibrations (Mem. Amer. Math. Soc. 155, 1975).Google Scholar
[McD1]McDuff, D.. Configuration spaces of positive and negative particles. Topology 14 (1975), 91–107.CrossRefGoogle Scholar
[McD2]McDuff, D.. Foliations and monoids of embeddings, in Geometric Topology, Ed. Cantrell, (Academic Press, 1979), 429–444.CrossRefGoogle Scholar
[McD3]McDuff, D.. The homology of some groups of diffeomorphisms. Comment. Math. Helv. 55 (1980), 97–129.CrossRefGoogle Scholar
[McD–Se]McDuff, D. and Segal, G.. Homology fibrations and the group completion theorem. Invent. Math. 31 (1976), 279–284.CrossRefGoogle Scholar
[Qu]Quinn, F.. Topological transversality holds in all dimensions. Bull A.M.S. 18 (1988), 145–148.CrossRefGoogle Scholar
[Se1]Segal, G.. Configuration spaces and Iterated loop spaces. Invent. Math. 21 (1973), 213–221.CrossRefGoogle Scholar
[Se2]Segal, G.. Categories and cohomology theories. Topology 13 (1974), 293–312.CrossRefGoogle Scholar
[St]Stong, R.. Notes on cobordism theory (P.U. Press, 1968).Google Scholar
[Th]Thomason, R. W.. Homotopy colimits in the category of small categories. Math. Proc. Cambridge Phil. Soc. 85 (1979), 91–109.CrossRefGoogle Scholar
[Ts1]Tsuboi, T.. Rationality of piecewise linear foliations and homology of the group of piecewise linear homeomorphisms. L'Enseignement Math. 38 (1992), 329–344.Google Scholar
[Ts2]Tsuboi, T.. Γ1-structures avec une seule feuille. Astérisque 116 (1984), 222–234.Google Scholar