Toward the group completion of the Burau representation

Morava, Jack; Rolfsen, Dale

doi:10.1090/tran/8796

Toward the group completion of the Burau representation
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Trans. Amer. Math. Soc. 376 (2023), 1845-1865 Request permission

Abstract:

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an $E_2$ Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of $\mathbb {C}$, and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on $\mathbb {R}^3_+$.

References

Omar Antolín-Camarena and Tobias Barthel, A simple universal property of Thom ring spectra, J. Topol. 12 (2019), no. 1, 56–78. MR 3875978, DOI 10.1112/topo.12084
Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, and Charles Rezk, An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology, J. Topol. 7 (2014), no. 3, 869–893. MR 3252967, DOI 10.1112/jtopol/jtt035
Jonathan Beardsley, Relative Thom spectra via operadic Kan extensions, Algebr. Geom. Topol. 17 (2017), no. 2, 1151–1162. MR 3623685, DOI 10.2140/agt.2017.17.1151
Jonathan Beardsley and Jack Morava, Toward a Galois theory of the integers over the sphere spectrum, J. Geom. Phys. 131 (2018), 41–51. MR 3815226, DOI 10.1016/j.geomphys.2018.04.011
Agnès Beaudry, Paul G. Goerss, Michael J. Hopkins, and Vesna Stojanoska, Dualizing spheres for compact $p$-adic analytic groups and duality in chromatic homotopy, Invent. Math. 229 (2022), no. 3, 1301–1434. MR 4462627, DOI 10.1007/s00222-022-01120-1
Julia E. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2043–2058. MR 2276611, DOI 10.1090/S0002-9947-06-03987-0
Joan S. Birman and Tara E. Brendle, Braids: a survey, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 19–103. MR 2179260, DOI 10.1016/B978-044451452-3/50003-4
J. S. Birman, D. D. Long, and J. A. Moody, Finite-dimensional representations of Artin’s braid group, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992) Contemp. Math., vol. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 123–132. MR 1292900, DOI 10.1090/conm/169/01655
Andrew J. Blumberg, Topological Hochschild homology of Thom spectra which are $E_\infty$ ring spectra, J. Topol. 3 (2010), no. 3, 535–560. MR 2684512, DOI 10.1112/jtopol/jtq017
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609, DOI 10.1007/BFb0068547
A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), no. 6, 1321–1354. MR 1481817, DOI 10.1353/ajm.1997.0036
J Brundan, Representations of the oriented skein category, https://arxiv.org/abs/1712.08953
J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. of Math. (2) 143 (1996), no. 3, 435–467. MR 1394965, DOI 10.2307/2118532
Werner Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 179–186 (German). MR 3069652, DOI 10.1007/BF02940722
Filippo Callegaro, The homology of the Milnor fiber for classical braid groups, Algebr. Geom. Topol. 6 (2006), 1903–1923. MR 2263054, DOI 10.2140/agt.2006.6.1903
P. Carrasco, A. M. Cegarra, and A. R. Garzón, Classifying spaces for braided monoidal categories and lax diagrams of bicategories, Adv. Math. 226 (2011), no. 1, 419–483. MR 2735766, DOI 10.1016/j.aim.2010.06.027
Thomas Cauwbergs, Logarithmic geometry and the Milnor fibration, C. R. Math. Acad. Sci. Paris 354 (2016), no. 7, 701–706 (English, with English and French summaries). MR 3508567, DOI 10.1016/j.crma.2016.04.005
D Clausen, $p$-adic $J$-homomorphisms and a product formula (§4.4 p 12), arXiv:1110.5851
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, DOI 10.1007/BFb0080464
F. R. Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978), no. 2, 99–110. MR 493954, DOI 10.1007/BF01393249
F. R. Cohen, J. P. May, and L. R. Taylor, $K(\textbf {Z},\,0)$ and $K(Z_{2},\,0)$ as Thom spectra, Illinois J. Math. 25 (1981), no. 1, 99–106. MR 602900
F. R. Cohen and J. Pakianathan, The stable braid group and the determinant of the Burau representation, Proceedings of the Nishida Fest (Kinosaki 2003), Geom. Topol. Monogr., vol. 10, Geom. Topol. Publ., Coventry, 2007, pp. 117–129. MR 2402779, DOI 10.2140/gtm.2007.10.117
Patrick Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), no. 1, 115–150. MR 1214782, DOI 10.1090/S0002-9947-1994-1214782-4
Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, Mathematical Surveys and Monographs, vol. 148, American Mathematical Society, Providence, RI, 2008. MR 2463428, DOI 10.1090/surv/148
G. Denham and N. Lemire, Equivariant Euler characteristics of discriminants of reflection groups, Indag. Math. (N.S.) 13 (2002), no. 4, 441–458. MR 2015830, DOI 10.1016/S0019-3577(02)80025-8
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Edward R. Fadell and Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. MR 1802644, DOI 10.1007/978-3-642-56446-8
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest, Ordering the braid groups, Pacific J. Math. 191 (1999), no. 1, 49–74. MR 1725462, DOI 10.2140/pjm.1999.191.49
Peter J. Freyd and David N. Yetter, Braided compact closed categories with applications to low-dimensional topology, Adv. Math. 77 (1989), no. 2, 156–182. MR 1020583, DOI 10.1016/0001-8708(89)90018-2
E. A. Gorin and V. Ja. Lin, Algebraic equations with continuous coefficients, and certain questions of the algebraic theory of braids, Mat. Sb. (N.S.) 78 (120) (1969), 579–610 (Russian). MR 0251712
Amrani Ilias, Model structure on the category of small topological categories, J. Homotopy Relat. Struct. 10 (2015), no. 1, 63–70. MR 3313635, DOI 10.1007/s40062-013-0041-8
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI 10.1006/aima.1993.1055
André Joyal and Ross Street, Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), no. 1, 43–51. MR 1107651, DOI 10.1016/0022-4049(91)90039-5
André Joyal, Ross Street, and Dominic Verity, Traced monoidal categories, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 447–468. MR 1357057, DOI 10.1017/S0305004100074338
Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
Toshitake Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 339–363. MR 975088, DOI 10.1090/conm/078/975088
Lê Dũng Tráng, Some remarks on relative monodromy, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 397–403. MR 0476739
J Lurie, Poincaré spaces and Spivak fibrations, http://people.math.harvard.edu/~lurie/287xnotes/Lecture26.pdf
Ib Madsen and R. James Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, No. 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 548575
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, Douglas C. Ravenel, and Paul Shick, The Thomified Eilenberg-Moore spectral sequence, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 249–262. MR 1851257
S Majid, Anyonic groups, http://www.numdam.org/article/RCP25_1992__43__147_0.pdf
Zhouhang Mao, Perfectoid rings as Thom spectra, arXiv:2003.08697.
D. McDuff and G. Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1975/76), no. 3, 279–284. MR 402733, DOI 10.1007/BF01403148
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055
Thomas Nikolaus, Christoph Sachse, and Christoph Wockel, A smooth model for the string group, Int. Math. Res. Not. IMRN 16 (2013), 3678–3721. MR 3090706, DOI 10.1093/imrn/rns154
Franklin P. Peterson and Hirosi Toda, On the structure of $H^{\ast } (\textrm {BSF};\,Z_{p})$, J. Math. Kyoto Univ. 7 (1967), 113–121. MR 230316, DOI 10.1215/kjm/1250524271
Stewart Priddy, Lectures on the stable homotopy of $BG$, Proceedings of the School and Conference in Algebraic Topology, Geom. Topol. Monogr., vol. 11, Geom. Topol. Publ., Coventry, 2007, pp. 289–308. MR 2402810
Frank Quinn, Surgery on Poincaré and normal spaces, Bull. Amer. Math. Soc. 78 (1972), 262–267. MR 296955, DOI 10.1090/S0002-9904-1972-12950-1
Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), no. 4, 969–1014. MR 2219307, DOI 10.1090/S0894-0347-06-00521-2
J. Rognes, Galois extensions of structured ring spectra, AMS Memoirs, arXiv:0502183
Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393, DOI 10.1007/BF02684591
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
Mei Chee Shum, Tortile tensor categories, J. Pure Appl. Algebra 93 (1994), no. 1, 57–110. MR 1268782, DOI 10.1016/0022-4049(92)00039-T
Stephen Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc. 10 (1959), 621–626. MR 112149, DOI 10.1090/S0002-9939-1959-0112149-8
N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 210075, DOI 10.1307/mmj/1028999711
Dennis P. Sullivan, Geometric topology: localization, periodicity and Galois symmetry, $K$-Monographs in Mathematics, vol. 8, Springer, Dordrecht, 2005. The 1970 MIT notes; Edited and with a preface by Andrew Ranicki. MR 2162361, DOI 10.1007/1-4020-3512-8
Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/gsm/145
Atsushi Yamaguchi, Morava $K$-theory of double loop spaces of spheres, Math. Z. 199 (1988), no. 4, 511–523. MR 968317, DOI 10.1007/BF01161640