On homological properties of singular braids
Author:
Vladimir V. Vershinin
Journal:
Trans. Amer. Math. Soc. 350 (1998), 2431-2455
MSC (1991):
Primary 20J05, 20F36, 20F38, 18D10, 55P35
DOI:
https://doi.org/10.1090/S0002-9947-98-02048-0
MathSciNet review:
1443895
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid
which embeds into the singular braid group
. The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group
where is an infinite loop space and
is a double loop space.
- [Ad] John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, vol. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 505692
- [Arn1] V. I. Arnol′d, Certain topological invariants of algebrac functions, Trudy Moskov. Mat. Obšč. 21 (1970), 27–46 (Russian). MR 0274462
- [Arn2] V. I. Arnol′d, Topological invariants of algebraic functions. II, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 1–9 (Russian). MR 0276244
- [Art1] E. Artin, Theorie der Zopfe, Abh. Math. Semin. Univ. Hamburg 4 (1925), 47-72.
- [Art2] E. Artin, Theory of braids, Ann. of Math. 48 (1947), 101-126. MR 8:367a
- [Bae] John C. Baez, Link invariants of finite type and perturbation theory, Lett. Math. Phys. 26 (1992), no. 1, 43–51. MR 1193625, https://doi.org/10.1007/BF00420517
- [Bar] 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
- [Bat] M. Batanin, Private communication.
- [Bi] Joan S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 253–287. MR 1191478, https://doi.org/10.1090/S0273-0979-1993-00389-6
- [CF1] Fred Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. 79 (1973), 763–766. MR 321074, https://doi.org/10.1090/S0002-9904-1973-13306-3
- [CF2] Fred Cohen, Homology of Ω⁽ⁿ⁺¹⁾Σ⁽ⁿ⁺¹⁾𝑋 and 𝐶_{(𝑛+1)}𝑋,𝑛>0, Bull. Amer. Math. Soc. 79 (1973), 1236–1241 (1974). MR 339176, https://doi.org/10.1090/S0002-9904-1973-13394-4
- [CF3] F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), no. 2, 99–110. MR 493954, https://doi.org/10.1007/BF01393249
- [CLM] 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
- [FKR] R. Fenn, E. Keyman and C. Rourke, The singular braid monoid embeds in a group, Preprint, 1996.
- [FRR1] 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
- [FRR2] Roger Fenn, Richárd Rimányi, and Colin Rourke, The braid-permutation group, Topology 36 (1997), no. 1, 123–135. MR 1410467, https://doi.org/10.1016/0040-9383(95)00072-0
- [Fi] Z. Fiedorowicz, Operads and iterated monoidal categories, Preprint (1995).
- [Fuks] D. B. Fuks, Quillenization and bordism, Funkcional. Anal. i Priložen. 8 (1974), no. 1, 36–42 (Russian). MR 0343301
- [H] Allen Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995), no. 1, 39–62. MR 1314940, https://doi.org/10.1007/BF02565999
- [J] 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, https://doi.org/10.2307/1971403
- [JS] André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, https://doi.org/10.1006/aima.1993.1055
- [ML] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
- [Mah1] Mark Mahowald, A new infinite family in ₂𝜋_{*}^{𝑠}, Topology 16 (1977), no. 3, 249–256. MR 445498, https://doi.org/10.1016/0040-9383(77)90005-2
- [Mah2] Mark Mahowald, Ring spectra which are Thom complexes, Duke Math. J. 46 (1979), no. 3, 549–559. MR 544245
- [May] J. P. May, 𝐸_{∞} spaces, group completions, and permutative categories, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London, 1974, pp. 61–93. London Math. Soc. Lecture Note Ser., No. 11. MR 0339152
- [P] Stewart B. Priddy, On Ω^{∞}𝑆^{∞} 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
- [S1] Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, https://doi.org/10.1007/BF01390197
- [S2] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, https://doi.org/10.1016/0040-9383(74)90022-6
- [V] 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
- [W] 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
Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 20J05, 20F36, 20F38, 18D10, 55P35
Retrieve articles in all journals with MSC (1991): 20J05, 20F36, 20F38, 18D10, 55P35
Additional Information
Vladimir V. Vershinin
Affiliation:
Institute of Mathematics, Novosibirsk, 630090, Russia
Email:
versh@math.nsc.ru
DOI:
https://doi.org/10.1090/S0002-9947-98-02048-0
Keywords:
Braid group,
permutation group,
homology,
classifying space,
loop space
Received by editor(s):
August 20, 1996
Article copyright:
© Copyright 1998
American Mathematical Society