On homological properties of singular braids

Vershinin, Vladimir

doi:10.1090/S0002-9947-98-02048-0

On homological properties of singular braids
HTML articles powered by AMS MathViewer

by Vladimir V. Vershinin PDF

Trans. Amer. Math. Soc. 350 (1998), 2431-2455 Request permission

Abstract:

Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev’s theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group $BP_{n}$ of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid $SB_{n}$ which embeds into the singular braid group $SG_{n}$. The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group \begin{equation*} \mathbb {Z}\times BBP_{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }\times S^{1} \times Y, \end{equation*} \begin{equation*} \mathbb {Z}\times BSG_{\infty }^{+}\simeq S^{1}\times \Omega ^{2} S^{2}\times W, \end{equation*} where $Y$ is an infinite loop space and $W$ is a double loop space.

References

John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 505692
V. I. Arnol′d, Certain topological invariants of algebrac functions, Trudy Moskov. Mat. Obšč. 21 (1970), 27–46 (Russian). MR 0274462
V. I. Arnol′d, Topological invariants of algebraic functions. II, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 1–9 (Russian). MR 0276244
E. Artin, Theorie der Zopfe, Abh. Math. Semin. Univ. Hamburg 4 (1925), 47–72.
A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
John C. Baez, Link invariants of finite type and perturbation theory, Lett. Math. Phys. 26 (1992), no. 1, 43–51. MR 1193625, DOI 10.1007/BF00420517
M. G. Barratt, A free group functor for stable homotopy, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R., 1971, pp. 31–35. MR 0324693
M. Batanin, Private communication.
Joan S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287. MR 1191478, DOI 10.1090/S0273-0979-1993-00389-6
Fred Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. 79 (1973), 763–766. MR 321074, DOI 10.1090/S0002-9904-1973-13306-3
Fred Cohen, Homology of $\Omega ^{(n+1)}\Sigma ^{(n+1)}X$ and $C_{(n+1)}X,\,n>0$, Bull. Amer. Math. Soc. 79 (1973), 1236–1241 (1974). MR 339176, DOI 10.1090/S0002-9904-1973-13394-4
F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), no. 2, 99–110. MR 493954, DOI 10.1007/BF01393249
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146
R. Fenn, E. Keyman and C. Rourke, The singular braid monoid embeds in a group, Preprint, 1996.
M. E. Bozhüyük (ed.), Topics in knot theory, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 399, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1257900, DOI 10.1007/978-94-011-1695-4
Roger Fenn, Richárd Rimányi, and Colin Rourke, The braid-permutation group, Topology 36 (1997), no. 1, 123–135. MR 1410467, DOI 10.1016/0040-9383(95)00072-0
Z. Fiedorowicz, Operads and iterated monoidal categories, Preprint (1995).
D. B. Fuks, Quillenization and bordism, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 36–42 (Russian). MR 0343301
Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), no. 1, 39–62. MR 1314940, DOI 10.1007/BF02565999
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
Mark Mahowald, A new infinite family in ${}_{2}\pi _{*}{}^s$, Topology 16 (1977), no. 3, 249–256. MR 445498, DOI 10.1016/0040-9383(77)90005-2
Mark Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), no. 3, 549–559. MR 544245
J. P. May, $E_{\infty }$ spaces, group completions, and permutative categories, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) London Math. Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974, pp. 61–93. MR 0339152
Stewart B. Priddy, On $\Omega ^{\infty }S^{\infty }$ and the infinite symmetric group, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 217–220. MR 0358767
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb. MR 1168473, DOI 10.1090/conm/478
Friedhelm Waldhausen, Algebraic $K$-theory of topological spaces. I, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 35–60. MR 520492