Relevant bibliographies by topics / Classifying spaces, homology, knots

Academic literature on the topic 'Classifying spaces, homology, knots'

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Published: 9 March 2023
Last updated: 10 March 2023

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

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Classifying spaces, homology, knots"

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PASINI, FEDERICO WILLIAM. "Classifying spaces for knots: new bridges between knot theory and algebraic number theory." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2016. http://hdl.handle.net/10281/129230.

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In this thesis we discuss how, in the context of knot theory, the classifying space of a knot group for the family of meridians arises naturally. We provide an explicit construction of a model for that space, which is particularly nice in the case of a prime knot. We then show that this classifying space controls the behaviour of the finite branched coverings of the knot. We present a 9-term exact sequence for knot groups that strongly resembles the Poitou-Tate exact sequence for algebraic number fields. Finally, we show that the homology of the classifying space behaves towards the former sequence as Shafarevich groups do towards the latter.
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Pelatt, Kristine, and Kristine Pelatt. "Geometry and Combinatorics Pertaining to the Homology of Spaces of Knots." Thesis, University of Oregon, 2012. http://hdl.handle.net/1794/12423.

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We produce explicit geometric representatives of non-trivial homology classes in Emb(S1,Rd), the space of knots, when d is even. We generalize results of Cattaneo, Cotta-Ramusino and Longoni to define cycles which live off of the vanishing line of a homology spectral sequence due to Sinha. We use con figuration space integrals to show our classes pair non-trivially with cohomology classes due to Longoni. We then give an alternate formula for the first differential in the homology spectral sequence due to Sinha. This differential connects the geometry of the cycles we define to the combinatorics of the spectral sequence. The new formula for the differential also simplifies calculations in the spectral sequence.
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Books on the topic "Classifying spaces, homology, knots"

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Moduli spaces of Riemann surfaces. Providence, Rhode Island: American Mathematical Society, 2013.

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Stanford Symposium on Algebraic Topology: Applications and New Directions (2012 : Stanford, Calif.), ed. Algebraic topology: Applications and new directions : Stanford Symposium on Algebraic Topology: Applications and New Directions, July 23--27, 2012, Stanford University, Stanford, CA. Providence, Rhode Island: American Mathematical Society, 2014.

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editor, Donagi Ron, Katz Sheldon 1956 editor, Klemm Albrecht 1960 editor, and Morrison, David R., 1955- editor, eds. String-Math 2012: July 16-21, 2012, Universität Bonn, Bonn, Germany. Providence, Rhode Island: American Mathematical Society, 2015.

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The Ro(G)-Graded Equivariant Ordinary Homology of G-Cell Complexes With Even-Dimensional Cells for G=Z/P (Memoirs of the American Mathematical Society). American Mathematical Society, 2004.

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Ellis, Graham. An Invitation to Computational Homotopy. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198832973.001.0001.

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This book is an introduction to elementary algebraic topology for students with an interest in computers and computer programming. Its aim is to illustrate how the basics of the subject can be implemented on a computer. The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues and it is hoped that the treatment of these will also appeal to readers already familiar with basic theory who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules, homotopy 2- types and explicit resolutions for an eclectic selection of discrete groups. It attempts to cover these topics in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject. The applied examples in the initial chapters use only low-dimensional and mainly abelian topological tools. Our applications of higher dimensional and less abelian computational methods are currently confined to pure mathematical calculations. The approach taken to computational homotopy is very much based on J.H.C. Whitehead’s theory of combinatorial homotopy in which he introduced the fundamental notions of CW-space, simple homotopy equivalence and crossed module. The book should serve as a self-contained informal introduction to these topics and their computer implementation. It is written in a style that tries to lead as quickly as possible to a range of potentially useful machine computations.
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Book chapters on the topic "Classifying spaces, homology, knots"

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Dwyer, W. G. "Classifying Spaces and Homology Decompositions." In Homotopy Theoretic Methods in Group Cohomology, 1–53. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8356-6_1.

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Haesemeyer, Christian, and Charles A. Weibel. "Motivic Classifying Spaces." In The Norm Residue Theorem in Motivic Cohomology, 253–76. Princeton University Press, 2019. http://dx.doi.org/10.23943/princeton/9780691191041.003.0015.

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This chapter focuses on motivic classifying spaces. It first connects the motives 𝑆∞ tr(𝕃𝑛) to cohomology operations on 𝐻2𝑛, 𝑛, at least when char(𝑘)=0. This parallels the Dold–Thom theorem in topology, which identifies the reduced homology ̃𝐻*(𝑋, ℤ) of a connected space 𝑋 with the homotopy groups of the infinite symmetric product 𝑆∞𝑋. A similar analysis shows that 𝔾𝑚 represents 𝐻1,1(−, ℤ), which allows us to describe operations on 𝐻1,1. The chapter then introduces the notion of scalar weight operations on 𝐻2𝑛, 𝑛. Afterward, it develops formulas for 𝑆𝓁tr(𝕃𝑛). These formulas imply that 𝑆∞ tr(𝕃𝑛) is a proper Tate motive, so there is a Künneth formula for them. The chapter culminates in a theorem demonstrating that β‎𝑃𝑏 is the unique cohomology operation of scalar weight 0 in its bidegree.
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Conference papers on the topic "Classifying spaces, homology, knots"

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Sakai, Keiichi. "Poisson structures on the homology of the space of knots." In Groups, homotopy and configuration spaces, in honour of Fred Cohen's 60th birthday. Mathematical Sciences Publishers, 2008. http://dx.doi.org/10.2140/gtm.2008.13.463.

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Journal articles
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