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Journal articles on the topic 'Classifying spaces, homology, knots'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

VERSHININ, VLADIMIR V. "ON HOMOLOGY OF VIRTUAL BRAIDS AND BURAU REPRESENTATION." Journal of Knot Theory and Its Ramifications 10, no. 05 (2001): 795–812. http://dx.doi.org/10.1142/s0218216501001165.

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Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The group of virtual braids on n strings VBn and its Burau representation to GLnℤ[t,t-1] also can be considered. The homological properties of the series of groups VBn and its Burau representation are studied. The following splitting of infinite loop spaces is proved for the plus-construction of the classifying space of the virtual braid group on the infinite number of strings: [Formula: see text] where Y is an infinite loop space. Connections with K*ℤ are discussed.

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2

Aceto, Paolo, Daniele Celoria, and JungHwan Park. "Rational cobordisms and integral homology." Compositio Mathematica 156, no. 9 (2020): 1825–45. http://dx.doi.org/10.1112/s0010437x20007320.

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We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a con

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3

Vassiliev, V. A. "Homology of spaces of knots in any dimensions." Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 359, no. 1784 (2001): 1343–64. http://dx.doi.org/10.1098/rsta.2001.0838.

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4

KAWAUCHI, AKIO. "ON LINKING SIGNATURE INVARIANTS OF SURFACE-KNOTS." Journal of Knot Theory and Its Ramifications 11, no. 03 (2002): 369–85. http://dx.doi.org/10.1142/s0218216502001688.

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We show that the linking signature of a closed oriented 4-manifold with infinite cyclic first homology is twice the Rochlin invariant of an exact leaf with a spin support if such a leaf exists. In particular, the linking signature of a surface-knot in the 4-sphere is twice the Rochlin invariant of an exact leaf of an associated closed spin 4-manifold with infinite cyclic first homology. As an application, we characterize a difference between the spin structures on a homology quaternion space in terms of closed oriented 4-manifolds with infinite cyclic first homology, so that we can obtain exam

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5

Dembegioti, F., N. Petrosyan, and O. Talelli. "Intermediaries in Bredon (co)homology and classifying spaces." Publicacions Matemàtiques 56 (July 1, 2012): 393–412. http://dx.doi.org/10.5565/publmat_56212_06.

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6

Dwyer, W. G. "Homology decompositions for classifying spaces of finite groups." Topology 36, no. 4 (1997): 783–804. http://dx.doi.org/10.1016/s0040-9383(96)00031-6.

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7

Grandjean, A. R., M. Ladra, and T. Pirashvili. "CCG-Homology of Crossed Modules via Classifying Spaces." Journal of Algebra 229, no. 2 (2000): 660–65. http://dx.doi.org/10.1006/jabr.2000.8296.

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8

Clancy, Maura, and Graham Ellis. "Homology of some Artin and twisted Artin Groups." Journal of K-Theory 6, no. 1 (2009): 171–96. http://dx.doi.org/10.1017/is008008012jkt090.

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AbstractWe begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological calculations.

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9

OGASA, EIJI. "SUPERSYMMETRY, HOMOLOGY WITH TWISTED COEFFICIENTS AND n-DIMENSIONAL KNOTS." International Journal of Modern Physics A 21, no. 19n20 (2006): 4185–96. http://dx.doi.org/10.1142/s0217751x06030941.

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In this paper, we study and construct a set of Witten indexes for K, where K is any n-dimensional knot in Sn+2 and n is any natural number. We form a supersymmetric quantum system for K by, first, constructing a set of functional spaces (spaces of fermionic (resp. bosonic) states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Our Witten indexes are topological invariant and they are nonzero in general. These indexes are zero if K is equivalent to a trivial knot. Besides, our Witten indexes restrict to the Alexander polynomials of n-knots, and one of

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10

Joachim, Michael, and Wolfgang Lück. "TopologicalK–(co)homology of classifying spaces of discrete groups." Algebraic & Geometric Topology 13, no. 1 (2013): 1–34. http://dx.doi.org/10.2140/agt.2013.13.1.

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11

Strounine, Alexei. "Homology decompositions for classifying spaces of compact Lie groups." Transactions of the American Mathematical Society 352, no. 6 (2000): 2643–57. http://dx.doi.org/10.1090/s0002-9947-00-02427-2.

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12

Farsi, Carla, Laura Scull, and Jordan Watts. "Classifying spaces and Bredon (co)homology for transitive groupoids." Proceedings of the American Mathematical Society 148, no. 6 (2020): 2717–37. http://dx.doi.org/10.1090/proc/14930.

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13

CLAY, ADAM, and DALE ROLFSEN. "Ordered groups, eigenvalues, knots, surgery and L-spaces." Mathematical Proceedings of the Cambridge Philosophical Society 152, no. 1 (2011): 115–29. http://dx.doi.org/10.1017/s0305004111000557.

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AbstractWe establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a bi-ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard–Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a no

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14

Baker, Kenneth L. "The Poincaré homology sphere and almost-simple knots in lens spaces." Proceedings of the American Mathematical Society 142, no. 3 (2013): 1071–74. http://dx.doi.org/10.1090/s0002-9939-2013-11832-0.

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15

Beliakova, Anna, and Thang T. Q. Lê. "Integrality of quantum 3-manifold invariants and a rational surgery formula." Compositio Mathematica 143, no. 6 (2007): 1593–612. http://dx.doi.org/10.1112/s0010437x07003053.

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AbstractWe prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

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Songhafouo Tsopméné, Paul Arnaud, and Victor Turchin. "Rational homology and homotopy of high-dimensional string links." Forum Mathematicum 30, no. 5 (2018): 1209–35. http://dx.doi.org/10.1515/forum-2016-0192.

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AbstractArone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable i

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17

McGibbon, C. A. "Wilson spaces and stable splittings of BTr." Glasgow Mathematical Journal 36, no. 3 (1994): 287–90. http://dx.doi.org/10.1017/s0017089500030871.

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Let Q(X) denote and let BTr denote the classifying space of the r-torus. In [8], Segal showed that Q(BT1) is homotopy equivalent to a product BU × F where BU denotes the classifying space for stable complex vector bundles and F a space with finite homotopy groups. This result has been a very useful one. For example, in [5] it was used to show that up to a stable homotopy equivalence there is only one loop structure on the 3-sphere at each odd prime p. (The subsequent work of Dwyer, Miller, and Wilkerson shows this result is even true unstably, at every prime p.) In [6] it was used to classify,

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18

NOTBOHM, D. "HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS." Journal of the London Mathematical Society 64, no. 2 (2001): 472–88. http://dx.doi.org/10.1112/s0024610701002459.

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For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consi

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19

Lambrechts, Pascal, Victor Turchin, and Ismar Volić. "The rational homology of spaces of long knots in codimension>2." Geometry & Topology 14, no. 4 (2010): 2151–87. http://dx.doi.org/10.2140/gt.2010.14.2151.

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20

Christensen, Antje. "Homology of manifolds obtained by Dehn surgery on knots in lens spaces." Journal of Knot Theory and Its Ramifications 09, no. 04 (2000): 431–42. http://dx.doi.org/10.1142/s0218216500000219.

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The question whether or not a Dehn surgery on a knot in a lens space yields a lens space of the same order is investigated with homological techniques. Determining the first homology group of the lens space after surgery and of its covering yields some necessary conditions on the knot and the surgery curve. Application of these results along with a calculation of Seifert invariants answers the question completely for surgery on torus knots along nullhomological curves.

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21

Maruyama, Noriko. "A distribution of rational homology 3-spheres captured by the CWL invariant Phase 1." Journal of Knot Theory and Its Ramifications 26, no. 10 (2017): 1750054. http://dx.doi.org/10.1142/s0218216517500547.

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Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integ

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22

Arone, Gregory, and Victor Turchin. "On the rational homology of high-dimensional analogues of spaces of long knots." Geometry & Topology 18, no. 3 (2014): 1261–322. http://dx.doi.org/10.2140/gt.2014.18.1261.

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23

Yan, Dung Yung. "The Brown-Peterson homology of the classifying spaces BO and BO(n)." Journal of Pure and Applied Algebra 102, no. 2 (1995): 221–33. http://dx.doi.org/10.1016/0022-4049(94)00083-u.

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24

BODEN, HANS U., та CYNTHIA L. CURTIS. "THE SL2(ℂ) CASSON INVARIANT FOR SEIFERT FIBERED HOMOLOGY SPHERES AND SURGERIES ON TWIST KNOTS". Journal of Knot Theory and Its Ramifications 15, № 07 (2006): 813–37. http://dx.doi.org/10.1142/s0218216506004762.

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We derive a simple closed formula for the SL2(ℂ) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL2(ℂ) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL2(ℂ) Casson invariant.

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25

Kadokami, Teruhisa, and Yuichi Yamada. "A deformation of the Alexander polynomials of knots yielding lens spaces." Bulletin of the Australian Mathematical Society 75, no. 1 (2007): 75–89. http://dx.doi.org/10.1017/s0004972700038995.

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For a knot K in a homology 3-sphere Σ, by Σ(K;p/q), we denote the resulting 3-manifold of p/q-surgery along K. We say that the manifold or the surgery is of lens type if Σ(K;p/q) has the same Reidemeister torsion as a lens space.We prove that, for Σ(K;p/q) to be of lens type, it is a necessary and sufficient condition that the Alexander polynomial ΔK(t) of K is equal to that of an (i, j)-torus knot T(i, j) modulo (tp – 1).We also deduce two results: If Σ(K;p/q) has the same Reidemeister torsion as L(p, q') then (1) (2) The multiple of ΣK(tk) over k ∈ (i) is ±tm modulo (tp – 1), where (i) is th

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26

Liang, Shiyu. "Non-left-orderable surgeries on 1-bridge braids." Journal of Knot Theory and Its Ramifications 29, no. 12 (2020): 2050086. http://dx.doi.org/10.1142/s0218216520500868.

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Boyer, Gordon and Watson have conjectured that an irreducible rational homology [Formula: see text]-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in [Formula: see text] can produce large families of L-spaces, it is natural to examine the conjecture on these [Formula: see text]-manifolds. Greene, Lewallen and Vafaee have proved that all [Formula: see text]-bridge braids are L-space knots. In this paper, we consider three families of [Formula: see text]-bridge braids. First we calculate the knot groups and peripheral subgroups. We

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27

Maginnis, John, and Silvia Onofrei. "New Collections ofp-Subgroups and Homology Decompositions for Classifying Spaces of Finite Groups." Communications in Algebra 36, no. 7 (2008): 2466–80. http://dx.doi.org/10.1080/00927870802069795.

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28

TILLMANN, ULRIKE. "A splitting for the stable mapping class group." Mathematical Proceedings of the Cambridge Philosophical Society 127, no. 1 (1999): 55–65. http://dx.doi.org/10.1017/s0305004199003485.

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The main result of [15] is that the classifying space of the stable mapping class group after plus construction BΓ+∞ is an infinite loop space. This result is used to show that, localized away from two, a connected component of the stable homotopy groups of spheres QS0 splits off BΓ+∞. The splitting is a splitting of infinite loop spaces. It follows immediately that the homology with coefficients in ℤ[½] of the infinite symmetric group is a direct summand of the homology of the stable mapping class group.

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TILLMANN, ULRIKE. "Homology stability for symmetric diffeomorphism and mapping class groups." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (2015): 121–39. http://dx.doi.org/10.1017/s0305004115000638.

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AbstractFor any smooth compact manifold W with boundary of dimension of at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of k points or k embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of W connected sum with k copies of an arbitrary compact smooth manifold Q of the same dimension. The analogues for mapping class groups as well as other generalisations will also be proved.

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30

Audoux, Benjamin, and Delphine Moussard. "Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant." International Journal of Mathematics 30, no. 05 (2019): 1950021. http://dx.doi.org/10.1142/s0129167x19500216.

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In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined, respectively, by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degr

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31

GONZÁLEZ, JESÚS, MAURILIO VELASCO, and W. STEPHEN WILSON. "BIEQUIVARIANT MAPS ON SPHERES AND TOPOLOGICAL COMPLEXITY OF LENS SPACES." Communications in Contemporary Mathematics 15, no. 03 (2013): 1250051. http://dx.doi.org/10.1142/s0219199712500514.

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Weighted cup-length calculations in singular cohomology led Farber and Grant in 2008 to general lower bounds for the topological complexity of lens spaces. We replace singular cohomology by connective complex K-theory, and weighted cup-length arguments by considerations with biequivariant maps on spheres to improve on Farber–Grant's bounds by arbitrarily large amounts. Our calculations are based on the identification of key elements conjectured to generate the annihilator ideal of the toral bottom class in the ku-homology of the classifying space for a rank-2 abelian 2-group.

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32

MULAZZANI, MICHELE. "ALL LINS-MANDEL SPACES ARE BRANCHED CYCLIC COVERINGS OF S3." Journal of Knot Theory and Its Ramifications 05, no. 02 (1996): 239–63. http://dx.doi.org/10.1142/s0218216596000175.

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In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to co

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33

Songhafouo Tsopméné, Paul Arnaud. "Formality of Sinha’s cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology." Algebraic & Geometric Topology 13, no. 4 (2013): 2193–205. http://dx.doi.org/10.2140/agt.2013.13.2193.

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34

Baker, Andrew. "Husemoller-Witt decompositions and actions of the Steenrod algebra." Proceedings of the Edinburgh Mathematical Society 28, no. 2 (1985): 271–88. http://dx.doi.org/10.1017/s0013091500022690.

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Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], genera

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35

TOURTCHINE, V. "ON THE OTHER SIDE OF THE BIALGEBRA OF CHORD DIAGRAMS." Journal of Knot Theory and Its Ramifications 16, no. 05 (2007): 575–629. http://dx.doi.org/10.1142/s0218216507005397.

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In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in ℝd, d ≥ 3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in

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36

Arthamonov, S., and Sh Shakirov. "Refined Chern–Simons theory in genus two." Journal of Knot Theory and Its Ramifications 29, no. 07 (2020): 2050044. http://dx.doi.org/10.1142/s0218216520500443.

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Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus [Formula: see text] surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald [Formula: see text]-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus

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37

BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.

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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explana

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38

Hedden, Matthew, and Adam Simon Levine. "Splicing knot complements and bordered Floer homology." Journal für die reine und angewandte Mathematik (Crelles Journal) 2016, no. 720 (2016). http://dx.doi.org/10.1515/crelle-2014-0064.

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AbstractWe show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

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Kishimoto, Daisuke, and Stephen Theriault. "The mod-p homology of the classifying spaces of certain gauge groups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, September 19, 2022, 1–13. http://dx.doi.org/10.1017/prm.2022.61.

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Let $G$ be a compact connected simple Lie group of type $(n_{1},\,\ldots,\,n_{l})$ , where $n_{1}<\cdots < n_{l}$ . Let $\mathcal {G}_k$ be the gauge group of the principal $G$ -bundle over $S^{4}$ corresponding to $k\in \pi _3(G)\cong \mathbb {Z}$ . We calculate the mod- $p$ homology of the classifying space $B\mathcal {G}_k$ provided that $n_{l}< p-1$ .

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ZEMAN, TOMÁŠ. "On the quotients of mapping class groups of surfaces by the Johnson subgroups." Mathematical Proceedings of the Cambridge Philosophical Society, November 27, 2019, 1–23. http://dx.doi.org/10.1017/s0305004119000471.

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Abstract We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \time

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