Some remarks on configuration spaces

Raptis, George

doi:10.1090/S0002-9939-2010-10580-4

Some remarks on configuration spaces
HTML articles powered by AMS MathViewer

by George Raptis PDF

Proc. Amer. Math. Soc. 139 (2011), 1879-1887 Request permission

Abstract:

This paper studies the homotopy type of the configuration spaces $F_n(X)$ by introducing the idea of configuration spaces of maps. For every map $f: X \to Y$, the configuration space $F_n(f)$ is the space of configurations in $X$ that have distinct images in $Y$. We show that the natural maps $F_n(X) \leftarrow F_n(f) \rightarrow F_n(Y)$ are homotopy equivalences when $f$ is a proper cell-like map between $d$-manifolds. We also show that the best approximation to $X \mapsto F_n(X)$ by a homotopy invariant functor is given by the $n$-fold product map.

References

Mokhtar Aouina and John R. Klein, On the homotopy invariance of configuration spaces, Algebr. Geom. Topol. 4 (2004), 813–827. MR 2100681, DOI 10.2140/agt.2004.4.813
Steve Armentrout, Cellular decompositions of $3$-manifolds that yield $3$-manifolds, Memoirs of the American Mathematical Society, No. 107, American Mathematical Society, Providence, R.I., 1971. MR 0413104
C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 174–187. MR 922926, DOI 10.1007/BFb0078741
Morton Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74–76. MR 117695, DOI 10.1090/S0002-9904-1960-10400-4
T. A. Chapman, Cell-like mappings, Algebraic and geometrical methods in topology (Conf. Topological Methods in Algebraic Topology, State Univ. New York, Binghamton, N.Y., 1973), Lecture Notes in Math., Vol. 428, Springer, Berlin, 1974, pp. 230–240. MR 0383423
T. A. Chapman, Homotopy conditions which detect simple homotopy equivalences, Pacific J. Math. 80 (1979), no. 1, 13–46. MR 534693
Marshall M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 0362320
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Philip S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003. MR 1944041, DOI 10.1090/surv/099
Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
R. C. Lacher, Cell-like mappings of $\textrm {ANR}’s$, Bull. Amer. Math. Soc. 74 (1968), 933–935. MR 244963, DOI 10.1090/S0002-9904-1968-12093-2
R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717–731. MR 251714
Norman Levitt, Spaces of arcs and configuration spaces of manifolds, Topology 34 (1995), no. 1, 217–230. MR 1308497, DOI 10.1016/0040-9383(94)E0012-9
Riccardo Longoni and Paolo Salvatore, Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375–380. MR 2114713, DOI 10.1016/j.top.2004.11.002
W. J. R. Mitchell and D. Repovš, The topology of cell-like mappings, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), no. suppl., 265–300. Conference on Differential Geometry and Topology (Sardinia, 1988). MR 1122860
J. H. Roberts and N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. (2) 39 (1938), no. 4, 851–862. MR 1503441, DOI 10.2307/1968468
Frank Quinn, Ends of maps. III. Dimensions $4$ and $5$, J. Differential Geometry 17 (1982), no. 3, 503–521. MR 679069
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271–294. MR 295365, DOI 10.1016/0040-9383(72)90014-6
J. W. T. Youngs, Homeomorphic approximations to monotone mappings, Duke Math. J. 15 (1948), 87–94. MR 24623