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

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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 (August 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 (September 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 connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

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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 (July 15, 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 (May 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 examples showing that some different punctured embeddings into S4 produce different Rochlin invariants for some homology quaternion spaces.

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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 (July 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 (July 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 (September 21, 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 (August 10, 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 the Alexander polynomials of K is nontrivial if any of the Witten indexes is nonzero. Our indexes are related to homology with twisted coefficients. Roughly speaking, these indexes posseses path-integral representation in the usual manner of supersymmetric theory.

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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 (February 4, 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 (March 2, 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 (February 26, 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 (September 22, 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 nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

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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 (December 13, 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 (November 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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16

Songhafouo Tsopméné, Paul Arnaud, and Victor Turchin. "Rational homology and homotopy of high-dimensional string links." Forum Mathematicum 30, no. 5 (September 1, 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 in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.

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17

McGibbon, C. A. "Wilson spaces and stable splittings of BTr." Glasgow Mathematical Journal 36, no. 3 (September 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, up to homology, the stable self maps of the projective spaces ℂPn and ℍPn. In [5] I asked if a splitting similar to Segal's might exist for Q(BTr) when r≥2. In particular, since the homotopy and homology groups of BU are torsion free it seemed natural to ask if Q(BTr), when r>, could likewise contain a retract with torsion free homology and homotopy groups and whose complement is rationally trivial. The purpose of this note is to show that the answer is no.

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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 (October 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 consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

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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 (October 9, 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 (June 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 (September 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 integral surgeries along a knot [Formula: see text] with [Formula: see text].

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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 (July 7, 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 (July 1995): 221–33. http://dx.doi.org/10.1016/0022-4049(94)00083-u.

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24

BODEN, HANS U., and 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, no. 07 (September 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 (February 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 the subgroup in (Z/pZ)×/{±1} generated by i. Conversely, if a subgroup H of (Z/pZ)×/{±l} satisfying that the product of ΣK(tk) (k ∈ H) is ±tm modulo (tp – 1), then H includes i or j.Here, i, j are the parameters of the torus knot above.

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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 (October 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 then verify the conjecture on the three cases by applying the criterion developed by Christianson, Goluboff, Hamann and Varadaraj, when they verified the same conjecture for certain twisted torus knots and generalized the criteria due to Clay and Watson and due to Ichihara and Temma.

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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 (June 4, 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 (July 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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29

TILLMANN, ULRIKE. "Homology stability for symmetric diffeomorphism and mapping class groups." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 1 (December 2, 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 (May 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 degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of [Formula: see text]-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

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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 (May 19, 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 (April 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 contain all branched cyclic coverings of two-bridge knots or links.

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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 (June 6, 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 (June 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], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.

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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 (May 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 the discriminant (maps having only self-intersections). We interpret the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d - 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings, we introduce new homological operations on Hochschild complexes. We show in future work that these operations are in fact the Dyer–Lashof–Cohen operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

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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 (June 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 1 construction of arXiv:1105.5117 opening a question that if it extends to any genus. Our construction is built upon a [Formula: see text]-deformation of the square of [Formula: see text]-6[Formula: see text] symbol of [Formula: see text], which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.

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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 (August 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 explanation of the physical world, essentially depends on the possibility of building a new geometrical framework conceptually richer than Riemannian geometry. In fact, it still plays a fundamental role in non-Abelian gauge theories and in superstring theory, thanks to which a great variety of new mathematical structures has emerged. The scope of this presentation is to highlight the importance of these mathematical structures for theoretical physics. A very interesting hypothesis is that the global topological properties of the manifold's model of space–time play a major role in quantum field theory (QFT) and that, consequently, several physical quantum effects arise from the nonlocal changing metrical and topological structure of these manifold. Thus the unification of general relativity and quantum theory require some fundamental breakthrough in our understanding of the relationship between space–time and quantum process. In particular the superstring theories lead to the guess that the usual structure of space–time at the quantum scale must be dropped out from physical thought. Non-Abelian gauge theories satisfy the basic physical requirements pertaining to the symmetries of particle physics because they are geometric in character. They profoundly elucidate the fundamental role played by bundles, connections, and curvature in explaining the essential laws of nature. Kaluza–Klein theories and more remarkably superstring theory showed that space–time symmetries and internal (quantum) symmetries might be unified through the introduction of new structures of space with a different topology. This essentially means, in our view, that "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain class of (presumably) nonsmooth manifolds. In the second part of this paper, we address the subject of topological quantum field theories (TQFTs), which constitute a remarkably important meeting ground for physicists and mathematicians. TQFTs can be used as a powerful tool to probe geometry and topology in low dimensions. Chern–Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of QFTs which can be exactly (nonperturbatively) and explicitly solved. Abelian Chern–Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link (i.e. the union of a finite number of disjoint knots). In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the well-known Jones polynomial. Powerful methods for complete analytical and nonperturbative computation of these knot and link invariants have been developed. From these invariants for unoriented and framed links in S3, an invariant for any three-manifold can be easily constructed by exploiting the Lickorish–Wallace surgery presentation of three-manifolds. This invariant up to a normalization is the partition function of the Chern–Simons field theory. Even perturbative analysis of Chern–Simons theories are rich in their mathematical structure; these provide a field theoretic interpretation of Vassiliev knot invariants. In Donaldson–Witten theory perturbative methods have proved their relations to Donaldson invariants. Nonperturbative methods have been applied after the work by Seiberg and Witten on N = 2 supersymmetric Yang–Mills theory. The outcome of this application is a totally unexpected relation between Donaldson invariants and a new set of topological invariants called Seiberg–Witten invariants. Not only in mathematics, Chern–Simons theories find important applications in three- and four-dimensional quantum gravity also. Work on TQFT suggests that a quantum gravity theory can be formulated in three-dimensional space–time. Attempts have been made in the last years to formulate a theory of quantum gravity in four-dimensional space–time using "spin networks" and "spin foams". More generally, the developments of TQFTs represent a sort of renaissance in the relation between geometry and physics. The most important (new) feature of present developments is that links are being established between quantum physics and topology. Maybe this link essentially rests on the fact that both quantum theory and topology are characterized by discrete phenomena emerging from a continuous background. One very interesting example is the super-symmetric quantum mechanics theory, which has a deep geometric meaning. In the Witten super-symmetric quantum mechanics theory, where the Hamiltonian is just the Hodge–Laplacian (whereas the quantum Hamiltonian corresponding to a classical particle moving on a Riemannian manifold is just the Laplace–Beltrami differential operator), differential forms are bosons or fermions depending on the parity of their degrees. Witten went to introduce a modified Hodge–Laplacian, depending on a real-valued function f. He was then able to derive the Morse theory (relating critical points of f to the Betti numbers of the manifold) by using the standard limiting procedures relating the quantum and classical theories. Super-symmetric QFTs essentially should be viewed as the differential geometry of certain infinite-dimensional manifolds, including the associated analysis (e.g. Hodge theory) and topology (e.g. Betti numbers). A further comment is that the QFTs of interest are inherently nonlinear, but the nonlinearities have a natural origin, e.g. coming from non-Abelian Lie groups. Moreover there is usually some scaling or coupling parameter in the theory which in the limit relates to the classical theory. Fundamental topological aspects of such a quantum theory should be independent of the parameters and it is therefore reasonable to expect them to be computable (in some sense) by examining the classical limit. This means that such topological information is essentially robust and should be independent of the fine analytical details (and difficulties) of the full quantum theory. In the last decade much effort has been done to use these QFTs as a conceptual tool to suggest new mathematical results. In particular, they have led to spectacular progress in our understanding of geometry in low dimensions. It is most likely no accident that the usual QFTs can only be renormalized in (space–time) dimensions ≤4, and this is precisely the range in which difficult phenomena arise leading to deep and beautiful theories (e.g. the work of Thurston in three dimensions and Donaldson in four dimensions). It now seems clear that the way to investigate the subtleties of low-dimensional manifolds is to associate to them suitable infinite-dimensional manifolds (e.g. spaces of connections) and to study these by standard linear methods (homology, etc.). In other words we use QFT as a refined tool to study low-dimensional manifolds.

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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 (January 1, 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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39

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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40

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 ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

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