Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-13T08:38:06.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access Rights & Permissions [Opens in a new window]

Extract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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