Localization in Khovanov homology

Stoffregen, Matthew¹ ; Zhang, Melissa²

¹ Department of Mathematics, Michigan State University, East Lansing, MI, United States
² Department of Mathematics, University of California, Davis, Davis, CA, United States

Geometry & topology, Tome 28 (2024) no. 4, pp. 1501-1585.

Résumé

We construct equivariant Khovanov spectra for periodic links using the Burnside functor construction introduced by Lawson, Lipshitz, and Sarkar. By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in $S^3$.

Détail

DOI : 10.2140/gt.2024.28.1501

Classification : 57M25, 55P91
Keywords: Khovanov homology, periodic links, Smith inequality, localization, equivariant stable homotopy theory, Lipshitz–Sarkar Khovanov stable homotopy type, annular Khovanov homology, equivariant spectra, spectral sequence, low-dimensional topology, knot theory, odd Khovanov homology

Bibliographie

[1] M Abouzaid, I Smith, Khovanov homology from Floer cohomology, J. Amer. Math. Soc. 32 (2019) 1 | DOI

[2] J F Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, from: "Algebraic topology" (editors I Madsen, B Oliver), Lecture Notes in Math. 1051, Springer (1984) 483 | DOI

[3] R Akhmechet, V Krushkal, M Willis, Stable homotopy refinement of quantum annular homology, Compos. Math. 157 (2021) 710 | DOI

[4] R Akhmechet, V Krushkal, M Willis, Towards an sl2 action on the annular Khovanov spectrum, Adv. Math. 408 (2022) 108581 | DOI

[5] M M Asaeda, J H Przytycki, A S Sikora, Categorification of the skein module of tangles, from: "Primes and knots" (editors T Kohno, M Morishita), Contemp. Math. 416, Amer. Math. Soc. (2006) 1 | DOI

[6] D Auroux, J E Grigsby, S M Wehrli, Sutured Khovanov homology, Hochschild homology, and the Ozsváth–Szabó spectral sequence, Trans. Amer. Math. Soc. 367 (2015) 7103 | DOI

[7] D Bar-Natan, On Khovanov’s categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002) 337 | DOI

[8] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 | DOI

[9] A Beliakova, K K Putyra, S M Wehrli, Quantum link homology via trace functor, I, Invent. Math. 215 (2019) 383 | DOI

[10] J Bénabou, Introduction to bicategories, from: "Reports of the Midwest category seminar" (editors A Dold, B Eckmann), Lecture Notes in Math. 47, Springer (1967) 1 | DOI

[11] M Borodzik, W Politarczyk, M Silvero, Khovanov homotopy type, periodic links and localizations, Math. Ann. 380 (2021) 1233 | DOI

[12] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, 304, Springer (1972) | DOI

[13] K Boyle, Rank inequalities on knot Floer homology of periodic knots, Bull. Lond. Math. Soc. 54 (2022) 668 | DOI

[14] G E Bredon, Equivariant cohomology theories, Bull. Amer. Math. Soc. 73 (1967) 266 | DOI

[15] Y Chen, M Khovanov, An invariant of tangle cobordisms via subquotients of arc rings, Fund. Math. 225 (2014) 23 | DOI

[16] V Colin, P Ghiggini, K Honda, Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011) 8100 | DOI

[17] G Cooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978) 391 | DOI

[18] S R Costenoble, J P May, S Waner, Equivariant orientation theory, Homology Homotopy Appl. 3 (2001) 265 | DOI

[19] S R Costenoble, S Waner, Equivariant ordinary homology and cohomology, 2178, Springer (2016) | DOI

[20] E Dotto, K Moi, Homotopy theory of G–diagrams and equivariant excision, Algebr. Geom. Topol. 16 (2016) 325 | DOI

[21] W G Dwyer, D M Kan, J H Smith, Homotopy commutative diagrams and their realizations, J. Pure Appl. Algebra 57 (1989) 5 | DOI

[22] J P C Greenlees, J P May, Equivariant stable homotopy theory, from: "Handbook of algebraic topology" (editor I M James), North-Holland (1995) 277 | DOI

[23] J E Grigsby, A M Licata, S M Wehrli, Annular Khovanov homology and knotted Schur–Weyl representations, Compos. Math. 154 (2018) 459 | DOI

[24] J E Grigsby, S M Wehrli, An action of gl(1|1) on odd annular Khovanov homology, Math. Res. Lett. 27 (2020) 711 | DOI

[25] K Hendricks, A rank inequality for the knot Floer homology of double branched covers, Algebr. Geom. Topol. 12 (2012) 2127 | DOI

[26] K Hendricks, Localization of the link Floer homology of doubly-periodic knots, J. Symplectic Geom. 13 (2015) 545 | DOI

[27] K Hendricks, R Lipshitz, S Sarkar, A flexible construction of equivariant Floer homology and applications, J. Topol. 9 (2016) 1153 | DOI

[28] P Hu, D Kriz, I Kriz, Field theories, stable homotopy theory, and Khovanov homology, Topology Proc. 48 (2016) 327

[29] S Illman, Smooth equivariant triangulations of G–manifolds for G a finite group, Math. Ann. 233 (1978) 199 | DOI

[30] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 | DOI

[31] M Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665 | DOI

[32] M Khovanov, L Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008) 1 | DOI

[33] M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387 | DOI

[34] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM, I : Heegaard Floer homology and Seiberg–Witten Floer homology, Geom. Topol. 24 (2020) 2829 | DOI

[35] T Large, Equivariant Floer theory and double covers of three-manifolds, preprint (2019)

[36] A D Lauda, An introduction to diagrammatic algebra and categorified quantum sl2, Bull. Inst. Math. Acad. Sin. 7 (2012) 165

[37] T Lawson, R Lipshitz, S Sarkar, The cube and the Burnside category, from: "Categorification in geometry, topology, and physics" (editors A Beliakova, A D Lauda), Contemp. Math. 684, Amer. Math. Soc. (2017) 63 | DOI

[38] T Lawson, R Lipshitz, S Sarkar, Khovanov homotopy type, Burnside category and products, Geom. Topol. 24 (2020) 623 | DOI

[39] T Lawson, R Lipshitz, S Sarkar, Chen–Khovanov spectra for tangles, Michigan Math. J. 71 (2022) 401 | DOI

[40] T Lawson, R Lipshitz, S Sarkar, Khovanov spectra for tangles, J. Inst. Math. Jussieu 22 (2023) 1509 | DOI

[41] T Lidman, C Manolescu, The equivalence of two Seiberg–Witten Floer homologies, 399, Soc. Math. France (2018) | DOI

[42] T Lidman, C Manolescu, Floer homology and covering spaces, Geom. Topol. 22 (2018) 2817 | DOI

[43] R Lipshitz, S Sarkar, A Khovanov stable homotopy type, J. Amer. Math. Soc. 27 (2014) 983 | DOI

[44] R Lipshitz, D Treumann, Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers, J. Eur. Math. Soc. 18 (2016) 281 | DOI

[45] J P May, Equivariant homotopy and cohomology theory, 91, Amer. Math. Soc. (1996) | DOI

[46] J Musyt, Equivariant Khovanov homotopy type and periodic links, PhD thesis, University of Oregon (2019)

[47] P S Ozsváth, J Rasmussen, Z Szabó, Odd Khovanov homology, Algebr. Geom. Topol. 13 (2013) 1465 | DOI

[48] W Politarczyk, Equivariant Khovanov homology of periodic links, Michigan Math. J. 68 (2019) 859 | DOI

[49] L P Roberts, On knot Floer homology in double branched covers, Geom. Topol. 17 (2013) 413 | DOI

[50] S Sarkar, C Scaduto, M Stoffregen, An odd Khovanov homotopy type, Adv. Math. 367 (2020) 107112 | DOI

[51] S Sarkar, C Seed, Z Szabó, A perturbation of the geometric spectral sequence in Khovanov homology, Quantum Topol. 8 (2017) 571 | DOI

[52] P Seidel, I Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006) 453 | DOI

[53] P Seidel, I Smith, Localization for involutions in Floer cohomology, Geom. Funct. Anal. 20 (2010) 1464 | DOI

[54] P A Smith, Transformations of finite period, Ann. of Math. 39 (1938) 127 | DOI

[55] C Stroppel, Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (2005) 547 | DOI

[56] R M Vogt, Homotopy limits and colimits, Math. Z. 134 (1973) 11 | DOI

[57] M Willis, Stabilization of the Khovanov homotopy type of torus links, Int. Math. Res. Not. 2017 (2017) 3350 | DOI

[58] Y Xie, Instantons and annular Khovanov homology, Adv. Math. 388 (2021) 107864 | DOI

[59] M Zhang, A rank inequality for the annular Khovanov homology of 2–periodic links, Algebr. Geom. Topol. 18 (2018) 1147 | DOI