Dissertations / Theses: 'Homology theory'

1

Jacobsson, Magnus. "Khovanov homology and link cobordisms /." Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3765.

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2

McDougall, Adam Corey. "Relating Khovanov homology to a diagramless homology." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/709.

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A homology theory is defined for equivalence classes of links under isotopy in the 3-sphere. Chain modules for a link L are generated by certain surfaces whose boundary is L, using surface signature as the homological grading. In the end, the diagramless homology of a link is found to be equal to some number of copies of the Khovanov homology of that link. There is also a discussion of how one would generalize the diagramless homology theory (hence the theory of Khovanov homology) to links in arbitrary closed oriented 3-manifolds.

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3

Brautaset, Olav. "Homology Theory from the Geometric Viewpoint." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-15695.

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Given a multiplicative cohomology theory represented by a spectrum, E,we define its associated geometric homology theory by means of bordism. Restrictedto CW pairs, we show how the geometric homology theory is naturally equivalent to the homology theory associated E. This was done by M. Jakob in 2000, and we give an expositionfollowing his approach. We also consider a naturally occurring cap product.

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Yates, Stuart. "Discrete morse theory and L2 homology /." Title page, abstract and contents only, 1997. http://web4.library.adelaide.edu.au/theses/09SM/09smy34.pdf.

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5

Johnson, Christopher Aaron. "Applications of computational homology." Huntington, WV : [Marshall University Libraries], 2006. http://www.marshall.edu/etd/descript.asp?ref=621.

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6

Jia, Bei. "D-branes and K-homology." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/32039.

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In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,\phi]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $\phi: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,\phi]$ represents a D-brane configuration wrapping the subspace $\phi(M)$. As a consequence, the $K$-homology element defined by $[M,E,\phi]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with.
Master of Science

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7

Song, Yongjin. "Hermitian algebraic K-theory and dihedral homology /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487681788252481.

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8

Levikov, Filipp. "L-theory, K-theory and involutions." Thesis, University of Aberdeen, 2013. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=201918.

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In Part 1, we consider two descriptions of L-homology of a (polyhedron of a) simplicial complex X. The classical approach of Ranicki via (Z,X)-modules (cf. [Ran92]) iswell established and is used in Ranicki’s definition of the total surgery obstruction and his formulation of the algebraic surgery exact sequence (cf. [Ran79], [Ran92],[KMM]). This connection between algebraic surgery and geometric surgery has numerous applications in the theory of (highdimensional) manifolds. The approach described in [RW10] uses a category of homotopy complexes of cosheaves to construct for a manifold M a (rational) orientation class [M]L• in symmetric L-homology which is topologically invariant per construction. This is used to reprove the topological invariance of rational Pontryagin classes. The L-theory of the category of homotopy complexes of sheaves over an ENR X can be naturally identified with L-homology of X. If X is a simplicial complex, both definitions give L-homology, there is no direct comparison however. We close this gap by constructing a functor from the category of (Z,X)-modules to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss inducing an equivalence on L-theory. The work undertaken in Part 1 may be considered as an addendum to [RW10] and suggests some translation of ideas of [Ran92] into the language of [RW10]. Without significant alterations, this work may be generalised to the case of X being a △-set. The L-theory of △-sets is considered in [RW12]. Let A be a unital ring and I a category with objects given by natural numbers and two kinds of morphisms mn → n satisfying certain relations (see Ch.3.4). There is an I-diagram, given by n 7→ ˜K (A[x]/xn) where the tilde indicates the homotopy fiber of the projection induced map on algebraic K-theory (of free modules) K(A[x]/xn) → K(A). In Part 2 we consider the following result by Betley and Schlichtkrull [BS05]. After completion there is an equivalence of spectra TC(A)∧ ≃ holim I ˜K(A[x]/xn)∧ where TC(A) is the topological cyclic homology of A. This is a very important invariant of K-theory (cf. [BHM93], [DGM12]) and comes with the cyclotomic trace map tr : K(A) → TC(A). In [BS05], the authors prove that under the above identification the trace map corresponds to a “multiplication” with an element u∞ ∈ holim I ˜K (Z[x]/xn). In this work we are interested in a generalisation of this result. We construct an element u∞ ∈ holim I ˜K(Cn). where Cn can be viewed as the category of freemodules over the nilpotent extension S[x]/xn of the sphere spectrum S. Let G be a discrete group and S[G] its spherical group ring. Using our lift of u∞ we construct a map trBS : K(S[G]) → holim I ˜K (CG n ) where CG n should be interpreted as the category of free modules over the extension S[G][x]/xn. After linearisation this map coincides with the trace map constructed by Betley and Schlichtkrull. We conjecture but do not prove, that after completion the domain coincides with the topological cyclic homology of S[G]. Some indication is given at the end of the final chapter. To construct the element u∞ we rely on a generalisation of a result of Grayson on the K-theory of endomorphisms (cf. [Gra77]). Denote by EndC the category of endomorphisms of finite CW-spectra and by RC the Waldhausen category of free CW-spectra with an action of N, which are finite in the equivariant sense. Cofibrations are given by cellular inclusions and weak equivalences are given bymaps inducing an equivalence of (reduced) cellular chain complexes of Z[x]-modules, after inverting the set {1 + xZ[x]}. In Chapter 5 we prove (5.8) that there is a homotopy equivalence of spectra ˜K (EndC) ≃ ˜K (RC). where tildes indicate that homotopy fibres of the respective projections are considered. Furthermore, we pursue the goal of constructing an involutive tracemap for themodel of [BS05]. We employ the framework ofWaldhausen categories with duality (cf. [WW98]) to introduce for any G involutions on holim I ˜K (CG n ). We give enough indication for our trace map being involutive, in particular in the last three sections of Chapter 5, we sketch how the generalisation of the theoremof Grayson (5.8) can be improved to an involutive version. In the final chapter, we develop this further. Assuming that the element u∞ ∈ holim I ˜K (Cn) is a homotopy fixed point of the introduced involution, we construct a map from quadratic L-theory of S[G] to the Tate homology spectrum of Z/2 acting on the fibre of trBS (see 6.9) : L•(S[G]) → (hofib(trBS))thZ/2 and discuss the connection of this to a conjecture of Rognes andWeiss. The two parts of the thesis are preluded with their own introduction andmay be read independently. The fewcross references are completely neglectible.

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Kaygun, Atabey. "Bialgebra cyclic homology with coefficients." Connect to this title online, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1107564231.

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Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains vii, 77 p.; also includes graphics Includes bibliographical references (p. 76-77). Available online via OhioLINK's ETD Center

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10

FrÃ¸yshov, Kim A. "On Floer homology and four-manifolds with boundary." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.282194.

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11

Cooper, Benjamin. "3-dimensional topological field theory and Harrison homology." Diss., [La Jolla] : University of California, San Diego, 2009. http://wwwlib.umi.com/cr/ucsd/fullcit?p3360059.

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Thesis (Ph. D.)--University of California, San Diego, 2009.
Title from first page of PDF file (viewed August 11, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 84-86).

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12

Cheng, Wing Kin. "Euler characteristic structure and weight homology /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?MATH%202004%20CHENGW.

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13

Tovstopyat-Nelip, Lev Igorevich. "Braids, transverse links and knot Floer homology:." Thesis, Boston College, 2019. http://hdl.handle.net/2345/bc-ir:108376.

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Thesis advisor: John A. Baldwin
Contact geometry has played a central role in many recent advances in low-dimensional topology; e.g. in showing that knot Floer homology detects the genus of a knot and whether a knot is fibered. It has also been used to show that the unknot, trefoil, and figure eight knot are determined by their Dehn surgeries. An important problem in 3-dimensional contact geometry is the classification of Legendrian and transverse knots. Such knots come equipped with some classical invariants. New invariants from knot Floer homology have been effective in distinguishing Legendrian and transverse knots with identical classical invariants, a notoriously difficult task. The Giroux correspondence allows contact structures to be studied via purely topological constructs called open book decompositions. Transverse links are then braids about these open books, which in turn may be thought of as mapping tori of diffeomorphisms of compact surfaces with boundary having marked points, which we refer to as pointed monodromies. In the first part of this thesis, we investigate properties of the transverse invariant in knot Floer homology, in particular its behavior for transverse closures of pointed monodromies possessing certain dynamical properties. The binding of an open book sits naturally as a transverse link in the supported contact manifold. We prove that the transverse link invariant in knot Floer homology of the binding union any braid about the open book is non-zero. As an application, we show that any pointed monodromy with fractional Dehn twist coefficient greater than one has non-zero transverse invariant, generalizing a result of Plamenevskaya for braids about the unknot. In the second part of this thesis, we define invariants of Legendrian and transverse links in universally tight lens spaces using grid diagrams, generalizing those defined by Ozsvath, Szabo and Thurston. We show that our invariants are equivalent to those defined by Lisca, Ozsvath, Szabo and Stipsicz for Legendrian and transverse links in arbitrary contact 3-manifolds. Our argument involves considering braids about rational open book decompositions and filtrations on knot Floer complexes
Thesis (PhD) — Boston College, 2019
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

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Grudskaya, Tatiana. "Computation of homology of low-dimensional spaces." Thesis, Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/28800.

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15

Thom, Andreas Berthold. "Connective E-theory and bivariant homology for C*-algebras." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=968501311.

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16

Lauda, Aaron Dean. "Open-closed topological quantum field theory and tangle homology." Thesis, University of Cambridge, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.614322.

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17

Boyd, Rachael. "Homology of Coxeter and Artin groups." Thesis, University of Aberdeen, 2018. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=237053.

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We calculate the second and third integral homology of arbitrary finite rank Coxeter groups. The first of these calculations refines a theorem of Howlett, the second is entirely new. We then prove that families of Artin monoids, which have the braid monoid as a submonoid, satisfy homological stability. When the K(π,1) conjecture holds this gives a homological stability result for the associated families of Artin groups. In particular, we recover a classic result of Arnol'd.

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18

Mattox, Wade. "Homology of Group Von Neumann Algebras." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/28397.

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In this paper the following conjecture is studied: the group von Neumann algebra N(G) is a flat CG-module if and only if the group G is locally virtually cyclic. This paper proves that if G is locally virtually cyclic, then N(G) is flat as a CG-module. The converse is proved for the class of all elementary amenable groups without infinite locally finite subgroups. Foundational cases for which the conjecture is shown to be true are the groups G=Z, G=ZxZ, G=Z*Z, Baumslag-Solitar groups, and some infinitely-presented variations of Baumslag-Solitar groups. Modules other than N(G), such as L^p-spaces and group C*-algebras, are considered as well. The primary tool that is used to achieve many of these results is group homology.
Ph. D.

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19

Lee, Ik Jae. "A new generalization of the Khovanov homology." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14170.

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Doctor of Philosophy
Department of Mathematics
David Yetter
In this paper we give a new generalization of the Khovanov homology. The construction begins with a Frobenius-algebra-like object in a category of graded vector-spaces with an anyonic braiding, with most of the relations weaken to hold only up to phase. The construction of Khovanov can be adapted to give a new link homology theory from such data. Both Khovanov's original theory and the odd Khovanov homology of Oszvath, Rassmusen and Szabo arise from special cases of the construction in which the braiding is a symmetry.

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Goedecke, Julia. "Three viewpoints on semi-abelian homology." Thesis, University of Cambridge, 2009. https://www.repository.cam.ac.uk/handle/1810/224397.

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The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.

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Mukherjee, Sujoy. "On Skein Modules and Homology Theories Related to Knot Theory." Thesis, The George Washington University, 2019. http://pqdtopen.proquest.com/#viewpdf?dispub=13810465.

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Classical knot theory is the study of simple closed curves in three dimensional space. The dissertation studies algebraic structures related to and motivated by knot theory. Since the first half of 1990, several homology theories have been defined for quandles and, more generally, for shelves whose axioms are motivated by the Reidemeister moves in knot theory. The second chapter of the dissertation studies these homology theories. For rack homology, one of the theorems assert that any finite Abelian group can appear as a torsion subgroup in the rack homology of some quandle. Furthermore, the second quandle homology of quasigroup Alexander quandles is expressed in terms of exterior algebras. Finally, the role of the associativity axiom in the category of self distributive algebraic structures is studied from the viewpoint of the homology theories. One of the results in this discussion explicitly computes the rack homology of shelves with a right fixed element.

Associative shelves form a very useful collection of algebraic structures when comparing the homology theories for shelves and semigroups. Examples of such homology theories include group homology, Hochschild homology, and multi term homology. However, from the perspective of self distributivity, rack and one term homology prove not to be very useful as demonstrated in Chapter two. The third chapter of the dissertation introduces a new homology theory to take care of this issue. In fact, this homology theory works for the family of the so-called quasibands which contain the collection of associative shelves. A theorem in this part of the dissertation computes the second homology group of finite semilattices which are also examples of quasibands. Moreover, the relation between this homology theory and Temperley-Lieb algebras is established.

The third part of the dissertation focuses on Khovanov homology. In particular, the main goal in this part is to study torsion in Khovanov homology. Z₂ torsion in Khovanov homology is much better understood than any other torsion. Three years ago, only a finite number of examples of knots and prime links with non Z₂ torsion in their Khovanov homology were known. It was also known that knots and links with thin Khovanov homology can contain only Z₂ torsion. The work on Khovanov homology in the dissertation introduces the first known examples of infinite families of knots and prime links with non Z₂ torsion in their Khovanov homology. More specifically, infinite families of knots and prime links containing Z₃, Z₄, Z₅, Z₇, and Z₈ torsion in their Khovanov homology are introduced. The aforementioned results assert that given some reasonably small torsion group, there are `many' examples of knots and links with the given torsion subgroup in their Khovanov homology. The latter half of this study addresses the `large' side, meaning, methods of finding knots and links with `large' torsion groups in their Khovanov homology. To this end, links with even torsion of order 2^s, for 0 < s < 24, in their Khovanov homology are introduced. In the case of odd order, knots and links with Z9, Z₂₅, and Z₂₇ torsion in their Khovanov homology are introduced. These results and discoveries are used to resolve all but one part of the Przytycki-Sazdanović braid conjecture.

The final part of the dissertation is motivated by the elegant product-to-sum formula of Frohman and Gelca for multiplying curves in the Kauffman bracket skein algebra of the thickened torus. After comparing the Kauffman bracket skein algebra of the thickened four-holed sphere with the Kauffman bracket skein algebra of the thickened torus, formulas for multiplying curves in some special families are discussed. Following this, an algorithm to multiply any two curves in the Kauffman bracket skein algebra of the thickened four-holed sphere is described.

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Masters, Joseph David. "Lengths and homology of hyperbolic 3-manifolds /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.

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23

Hubbard, Diana D. "Properties and applications of the annular filtration on Khovanov homology." Thesis, Boston College, 2016. http://hdl.handle.net/2345/bc-ir:106791.

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Thesis advisor: Julia E. Grigsby
The first part of this thesis is on properties of annular Khovanov homology. We prove a connection between the Euler characteristic of annular Khovanov homology and the classical Burau representation for closed braids. This yields a straightforward method for distinguishing, in some cases, the annular Khovanov homologies of two closed braids. As a corollary, we obtain the main result of the first project: that annular Khovanov homology is not invariant under a certain type of mutation on closed braids that we call axis-preserving. The second project is joint work with Adam Saltz. Plamenevskaya showed in 2006 that the homology class of a certain distinguished element in Khovanov homology is an invariant of transverse links. In this project we define an annular refinement of this element, kappa, and show that while kappa is not an invariant of transverse links, it is a conjugacy class invariant of braids. We first discuss examples that show that kappa is non-trivial. We then prove applications of kappa relating to braid stabilization and spectral sequences, and we prove that kappa provides a new solution to the word problem in the braid group. Finally, we discuss definitions and properties of kappa in the reduced setting
Thesis (PhD) — Boston College, 2016
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

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24

Durusoy, Daniel Selahi. "Heegaard Floer homology of certain 3-manifolds and cobordism invariants." Diss., Connect to online resource - MSU authorized users, 2008.

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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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Boily, Christian M. "Homological flows & star formation." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.321079.

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Heß, Alexander [Verfasser]. "Factorable Monoids : Resolutions and Homology via Discrete Morse Theory / Alexander Heß." Bonn : Universitäts- und Landesbibliothek Bonn, 2012. http://d-nb.info/1044081376/34.

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Hesselmann, Sabine. "Zur Torsion der Kohomologie S-arithmetischer Gruppen." Bonn : [s.n.], 1993. http://catalog.hathitrust.org/api/volumes/oclc/31482302.html.

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Adhikari, S. Prashanth. "Torsion in the homology of the general linear group for a ring of algebraic integers /." Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/5770.

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Harris, Julianne S. "On the mod 2 general linear group homology of totally real number rings /." Thesis, Connect to this title online; UW restricted, 1997. http://hdl.handle.net/1773/5812.

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31

Watson, Greg M. "Computation of homology and an application to the conley index." Diss., Georgia Institute of Technology, 1998. http://hdl.handle.net/1853/29916.

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32

Tram, Heather. "Khovanov Homology as an Generalization of the Jones Polynomial in Kauffman Terms." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1987.

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This paper explains the construction of Khovanov homology of which begins by un derstanding how Louis Kauﬀman generalizes the Jones polynomial using a state sum model of the bracket polynomial for an unoriented knot or link and in turn recovers the Jones polynomial, a knot invariant for an oriented knot or link. Kauﬀman associates the unknot by the polynomial (−A2 − A−2) whereas Khovanov associates the unknot by (q + q−1) through a change of variables. As an oriented knot or link K with n crossings produces 2n smoothings, Khovanov builds a commutative cube {0,1}n and associates a graded vector space to each smoothing in the cube. By deﬁning a diﬀerential operator on the directed edges of the cube so that adjacent states diﬀer by a type of smoothing for a ﬁxed cross ing, we can form chain groups which are direct sums of these vector spaces. Naturally we get a bi-graded (co)chain complex which is called the Khovanov complex. The resulting (co)homology groups of these (co)chains turns out to be invariant under the Reidemeister moves and taking the Euler characteristic of the Khovanov complex returns the very same Jones polynomial that we started with.

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33

Wienhard, Anna Katharina. "Bounded cohomology and geometry." Bonn : Mathematisches Institut der Universität, 2004. http://catalog.hathitrust.org/api/volumes/oclc/62768224.html.

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Weselmann, Uwe. "Kohomologische Kongruenzen zwischen automorphen Darstellungen von GL₂." Bonn : [s.n.], 1993. http://catalog.hathitrust.org/api/volumes/oclc/31445704.html.

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35

So, Bing-kwan. "On some examples of Poisson homology and cohomology analytic and lie theoretic approaches /." Click to view the E-thesis via HKUTO, 2005. http://sunzi.lib.hku.hk/hkuto/record/B3617063X.

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馮淑貞 and Suk-ching Fung. "Asymptotic vanishing theorem of cohomology groups on compact quotientsof the unit ball." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1998. http://hub.hku.hk/bib/B31220848.

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So, Bing-kwan, and 蘇鈵鈞. "On some examples of Poisson homology and cohomology: analytic and lie theoretic approaches." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2005. http://hub.hku.hk/bib/B3617063X.

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Pearson, Kelly Jeanne. "Cohomology of the Orlik-Solomon algebras." [Eugene, Or. : University of Oregon Library System], 2000. http://libweb.uoregon.edu/UOTheses/2000/pearsonk00.pdf.

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Fung, Suk-ching. "Asymptotic vanishing theorem of cohomology groups on compact quotients of the unit ball /." Hong Kong : University of Hong Kong, 1998. http://sunzi.lib.hku.hk/hkuto/record.jsp?B20667991.

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Antal, Tamás. "Cyclic cohomological computations for the Connes-Moscovici-Kreimer Hopf algebras." Connect to this title online, 2004. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1092777186.

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Kahle, Matthew. "Topology of random simplicial complexes and phase transitions for homology /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5809.

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42

Rüschoff, Christian [Verfasser], and Otmar [Akademischer Betreuer] Venjakob. "Relative algebraic K-theory and algebraic cyclic homology / Christian Rüschoff ; Betreuer: Otmar Venjakob." Heidelberg : Universitätsbibliothek Heidelberg, 2016. http://d-nb.info/1180737873/34.

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43

Boquet, Grant Michael. "Multidimensional Behavioral Complexes." Thesis, Virginia Tech, 2008. http://hdl.handle.net/10919/31528.

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In a preprint by J. Wood, V. Lomadze, and E. Rogers, chains and boundary maps were defined for 2-D discrete behavioral systems. The corresponding homology groups were studied and tied to trajectory properties. Indeed, the homology groups encapsulated the concepts of autonomy, controllability, and signal restriction. We shall present an extension of their work to n-D discrete behavioral systems. In particular, we shall streamline the construction of the chain groups, the boundary maps between chains, and the study of the resultant homology groups. While constructing this machinery, we shall point out intrinsic flaws in their approach that make extension of their results less systematic. Finishing remarks shall be made on using the homology groups to determine system properties and potentially classify forms of controllability.
Master of Science

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44

Heistercamp, Muriel. "The Weinstein conjecture with multiplicities on spherizations." Doctoral thesis, Universite Libre de Bruxelles, 2011. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209882.

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Soit M une variété lisse fermée et considérons sont fibré cotangent T*M muni de la structure symplectique usuelle induite par la forme de Liouville. Une hypersurface S de T*M$ est dite étoilée fibre par fibre si pour tout point q de M, l'intersection Sq de S avec la fibre au dessus de q est le bord d'un domaine étoilé par rapport à l'origine 0q de la fibre T*qM. Un flot est naturellement associé à S, il s'agit de l'unique flot généré par le champ de Reeb le long de S, le flot de Reeb.

L'existence d'une orbite orbite fermée du flot de Reeb sur S fut annoncée par Weinstein dans sa conjecture en 1978. Indépendamment, Weinstein et Rabinowitz ont montré l'existence d'une orbite fermée sur les hypersurfaces de type étoilées dans l'espace réel de dimension 2n. Sous les hypothèses précédentes, l'existence d'une orbite fermée fut démontrée par Hofer et Viterbo. Dans le cas particulier du flot géodésique, l'existence de plusieurs orbites fermées fut notamment étudiée par Gromov, Paternain et Paternain-Petean. Dans cette thèse, ces résultats sont généralisés.

Les résultats principaux de cette thèse montrent que la structure topologique de la variété M implique, pour toute hypersurface étoilée fibre par fibre, l'existence de beaucoup d'orbites fermées du flot de Reeb. Plus précisément, une borne inférieure de la croissance du nombre d'orbites fermées du flot de Reeb en fonction de leur période est mise en évidence. /

Let M be a smooth closed manifold and denote by T*M the cotangent bundle over M endowed with its usual symplectic structure induced by the Liouville form. A hypersurface S of T*M is said to be fiberwise starshaped if for each point q in M the intersection Sq of S with the fiber at q bounds a domain starshaped with respect to the origin 0q in T*qM. There is a flow naturally associated to S, generated by the unique Reeb vector field R along S ,the Reeb flow.

The existence of one closed orbit was conjectured by Weinstein in 1978 in a more general setting. Independently, Weinstein and Rabinowitz established the existence of a closed orbit on star-like hypersurfaces in the 2n-dimensional real space. In our setting the Weinstein conjecture without the assumption was proved in 1988 by Hofer and Viterbo. The existence of many closed orbits has already been well studied in the special case of the geodesic flow, for example by Gromov, Paternain and Paternain-Petean. In this thesis we will generalize their results.

The main result of this thesis is to prove that the topological structure of $M$ forces, for all fiberwise starshaped hypersurfaces S, the existence of many closed orbits of the Reeb flow on S. More precisely, we shall give a lower bound of the growth rate of the number of closed Reeb-orbits in terms of their periods.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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45

Kühnlein, Stefan. "Kohomologie spezieller S-arithmetischer Gruppen und Modulformen." Bonn : [s.n.], 1994. http://catalog.hathitrust.org/api/volumes/oclc/31760587.html.

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46

Holmstrom, Andreas. "Arakelov motivic cohomology." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.607895.

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47

Membrillo-Hernandez, Fausto Humberto. "Homological properties of finite-dimensional algebras." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.670284.

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48

Knapp, Adam C. "Computations of Floer homology and gauge theoretic invariants for Montesinos twins." Diss., Connect to online resource - MSU authorized users, 2008.

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49

Agerberg, Jens. "Statistical Learning and Analysis on Homology-Based Features." Thesis, KTH, Matematisk statistik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-273581.

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Stable rank has recently been proposed as an invariant to encode the result of persistent homology, a method used in topological data analysis. In this thesis we develop methods for statistical analysis as well as machine learning methods based on stable rank. As stable rank may be viewed as a mapping to a Hilbert space, a kernel can be constructed from the inner product in this space. First, we investigate this kernel in the context of kernel learning methods such as support-vector machines. Next, using the theory of kernel embedding of probability distributions, we give a statistical treatment of the kernel by showing some of its properties and develop a two-sample hypothesis test based on the kernel. As an alternative approach, a mapping to a Euclidean space with learnable parameters can be conceived, serving as an input layer to a neural network. The developed methods are first evaluated on synthetic data. Then the two-sample hypothesis test is applied on the OASIS open access brain imaging dataset. Finally a graph classification task is performed on a dataset collected from Reddit.
Stable rank har föreslagits som en sammanfattning på datanivå av resultatet av persistent homology, en metod inom topologisk dataanalys. I detta examensarbete utvecklar vi metoder inom statistisk analys och maskininlärning baserade på stable rank. Eftersom stable rank kan ses som en avbildning i ett Hilbertrum kan en kärna konstrueras från inre produkten i detta rum. Först undersöker vi denna kärnas egenskaper när den används inom ramen för maskininlärningsmetoder som stödvektormaskin (SVM). Därefter, med grund i teorin för inbäddning av sannolikhetsfördelningar i reproducing kernel Hilbertrum, undersöker vi hur kärnan kan användas för att utveckla ett test för statistisk hypotesprövning. Slutligen, som ett alternativ till metoder baserade på kärnor, utvecklas en avbildning i ett euklidiskt rum med optimerbara parametrar, som kan användas som ett ingångslager i ett neuralt nätverk. Metoderna utvärderas först på syntetisk data. Vidare utförs ett statistiskt test på OASIS, ett öppet dataset inom neuroradiologi. Slutligen utvärderas metoderna på klassificering av grafer, baserat på ett dataset insamlat från Reddit.

QC 20200523

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50

Rada, Ion Kolster Manfred. "The Lichtenbaum conjecture at the prime 2 /." *McMaster only, 2002.

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Dissertations / Theses on the topic 'Homology theory'

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