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Dissertations / Theses on the topic 'Cohomology of condensed groups'

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Published: 14 December 2024

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1

Artusa, Marco. "Sur des théorèmes de dualité pour la cohomologie condensée du groupe de Weil d'un corps p-adique." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0228.

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L’objectif de cette thèse est double. Premièrement, on construit une théorie de cohomologie topologique pour le groupe de Weil d’un corps p-adique. En second lieu, on utilise cette théorie pour prouver des théorèmes de dualité, qui se manifestent sous la forme de la dualité de Pontryagin entre groupes abéliens localement compacts. Ces résultats améliorent des théorèmes de dualité existants et leur confèrent une perspective topologique. De tels objectifs peuvent être atteints grâce aux Mathématiques Condensées, qui fournissent un cadre dans lequel il est possible de faire de l’algèbre avec des objets topologiques. On définit une théorie cohomologique pour les groupes condensés et pro-condensés et on étudie ses propriétés. Ensuite, on applique cela au groupe de Weil d’un corps p-adique, considéré comme un groupe pro-condensé. On démontre que, dans certains cas particuliers, les groupes de cohomologie correspondants sont des groupes abéliens localement compacts de rangs finis. Ceci nous permet d’étendre la dualité locale de Tate à une catégorie plus générale de coefficients non nécessairement discrets, o`u elle prend la forme d’une dualité de Pontryagin entre groupes abéliens localement compacts. Dans la dernière partie de la thèse, on utilise le même cadre pour retrouver une version “à la Weil” de la dualité de Tate avec coefficients dans les variétés abéliennes, et plus généralement dans les 1- motifs, en exprimant ces dualités comme des accouplements parfaits entre groupes abéliens condensés. Pour ce faire, on associe à chaque groupe algébrique, resp. 1-motif, un groupe abélien condensé, resp. un complexe de groupes abéliens condensés, avec une action du groupe de Weil (pro-condensé). On appelle cette association la réalisation de Weil-étale condensée. On montre l’existence d’un accouplement de Poincaré condensé pour les variétés abéliennes, et on prouve une version condensée et “à la Weil” de la dualité de Tate à coefficients dans les variétés abéliennes, qui améliore le résultat correspondant de Karpuk. Enfin, on montre l’existence d’un accouplement de Poincaré condensé pour les 1-motifs. On prouve que cet accouplement est compatible à la filtration par les poids et on démontre un théorème de dualité à coefficients dans les 1- motifs, qui améliore un résultat de Harari-Szamuely
The goal of this thesis is twofold. First, we build a topological cohomology theory for the Weil group of p-adic fields. Secondly, we use this theory to prove duality theorems for such fields, which manifest as Pontryagin duality between locally compact abelian groups. These results improve existing duality theorems and give them a topological flavour. Condensed Mathematics allow us to reach these objectives, providing a framework where it is possible to do algebra with topological objects. We define and study a cohomology theory for condensed groups and pro-condensed groups, and we apply it to the Weil group of a p-adic field, considered as a pro-condensed group. The resulting cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups. In the last part of the thesis, we use the same framework to recover a Weil-version of the Tate duality with coefficients in abelian varieties and more generally in 1-motives, expressing those dualities as perfect pairings between condensed abelian groups. To do this, we associate to every algebraic group, resp. 1-motive, a condensed abelian group, resp. a complex of condensed abelian groups, with an action of the (pro-condensed) Weil group. We call this association the condensed Weil-´etale realisation. We show the existence of a condensed Poincar´e pairing for abelian varieties and we prove a condensed-Weil version of the Tate duality with coefficients in abelian varieties, which improves the correspondent result of Karpuk. Lastly, we exhibit a condensed Poincar´e pairing for 1-motives. We show that this pairing is compatible with the weight filtration and we prove a duality theorem with coefficients in 1-motives, which improves a result of Harari-Szamuely
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2

Watson, Toni Aliza. "Twisted cohomology groups." College Park, Md. : University of Maryland, 2006. http://hdl.handle.net/1903/3929.

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Thesis (M.A.) -- University of Maryland, College Park, 2006.
Thesis research directed by: Dept. of Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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3

Česnavičius, Kęstutis. "Selmer groups as flat cohomology groups." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/90180.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 44-46).
Given a prime number p, Bloch and Kato showed how the p Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm1-Selmer group Selpmn A need not be determined by the mod pm Galois representation A[pm]; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes E depending on K and A, such that Selpm A is determined by A[pm] for all ... In the course of the argument we describe the flat cohomology group ... of the ring of integers of K with coefficients in the pm- torsion A[pm] of the Neron model of A by local conditions for p V E, compare them with the local conditions defining Selm 2A, and prove that A[p't ] itself is determined by A[pm] for such p. Our method sharpens the relationship between Selpm A and ... which was observed by Mazur and continues to work for other isogenies 0 between abelian varieties over global fields provided that deg o is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve 11A1 over certain families of number fields. Standard glueing techniques developed in the course of the proofs have applications to finite flat group schemes over global bases, permitting us to transfer many of the known local results to the global setting.
by Kęstutis Česnavičius.
Ph. D.
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4

Clark, Jonathan Owen. "Cohomology of some finite groups." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240535.

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5

Eastridge, Samuel Vance. "First l^2-Cohomology Groups." Thesis, Virginia Tech, 2015. http://hdl.handle.net/10919/52952.

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We want to take a look at the first cohomology group H^1(G, l^2(G)), in particular when G is locally-finite. First, though, we discuss some results about the space H^1(G, C G) for G locally-finite, as well as the space H^1(G, l^2(G)) when G is finitely generated. We show that, although in the case when G is finitely generated the embedding of C G into l^2(G) induces an embedding of the cohomology groups H^1(G, C G) into H^1(G, l^2(G)), when G is countably-infinite locally-finite, the induced homomorphism is not an embedding. However, even though the induced homomorphism is not an embedding, we still have that H^1(G, l^2(G)) neq 0 when G is countably-infinite locally-finite. Finally, we give some sufficient conditions for H^1(G,l^2(G)) to be zero or non-zero.
Master of Science
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6

QUADRELLI, CLAUDIO. "Cohomology of Absolute Galois Groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/56993.

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The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-p case, i.e., one would like to know which pro-p groups occur as maximal pro-p Galois groups, i.e., maximal pro-p quotients of absolute Galois groups. Indeed, pro-p groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-p groups, called Bloch-Kato pro-p group, whose Galois cohomology satisfies the consequences of the Bloch-Kato conjecture. Also we introduce the notion of cyclotomic orientation for a pro-p group. With this approach, we are able to recover new substantial information about the structure of maximal pro-p Galois groups, and in particular on theta-abelian pro-p groups, which represent the "upper bound" of such groups. Also, we study the restricted Lie algebra and the universal envelope induced by the Zassenhaus filtration of a maximal pro-p Galois group, and their relations with Galois cohomology via Koszul duality. Altogether, this thesis provides a rather new approach to maximal pro-p Galois groups, besides new substantial results.
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7

Leary, Ian James. "The cohomology of certain finite groups." Thesis, University of Cambridge, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386114.

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8

Kim, Yunhyong. "Smooth cochain cohomology of loop groups." Thesis, University of Cambridge, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621575.

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9

Foster-Greenwood, Briana A. "Hochschild Cohomology and Complex Reflection Groups." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc149591/.

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A concrete description of Hochschild cohomology is the first step toward exploring associative deformations of algebras. In this dissertation, deformation theory, geometry, combinatorics, invariant theory, representation theory, and homological algebra merge in an investigation of Hochschild cohomology of skew group algebras arising from complex reflection groups. Given a linear action of a finite group on a finite dimensional vector space, the skew group algebra under consideration is the semi-direct product of the group with a polynomial ring on the vector space. Each representation of a group defines a different skew group algebra, which may have its own interesting deformations. In this work, we explicitly describe all graded Hecke algebras arising as deformations of the skew group algebra of any finite group acting by the regular representation. We then focus on rank two exceptional complex reflection groups acting by any irreducible representation. We consider in-depth the reflection representation and a nonfaithful rotation representation. Alongside our study of cohomology for the rotation representation, we develop techniques valid for arbitrary finite groups acting by a representation with a central kernel. Additionally, we consider combinatorial questions about reflection length and codimension orderings on complex reflection groups. We give algorithms using character theory to compute reflection length, atoms, and poset relations. Using a mixture of theory, explicit examples, and calculations using the software GAP, we show that Coxeter groups and the infinite family G(m,1,n) are the only irreducible complex reflection groups for which the reflection length and codimension orders coincide. We describe the atoms in the codimension order for the groups G(m,p,n). For arbitrary finite groups, we show that the codimension atoms are contained in the support of every generating set for cohomology, thus yielding information about the degrees of generators for cohomology.
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Anwar, Muhammad F. "Representations and cohomology of algebraic groups." Thesis, University of York, 2011. http://etheses.whiterose.ac.uk/2032/.

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Let G be a semisimple simply connected linear algebraic group over an algebraically closed field k of characteristic p. In [11], Donkin gave a recursive description for the characters of cohomology of line bundles on the flag variety G/B with G = SL3. In chapter 2 of this thesis we try to give a non recursive description for these characters. In chapter 3, we give the first step of a version of formulae in [11] for G = G2. In his famous paper [7], Demazure introduced certain indecomposable modules and used them to give a short proof of the Borel-Weil-Bott theorem (characteristic zero). In chapter 5 we give the cohomology of these modules. In a recent paper [17], Doty introduces the notion of r−minuscule weight and exhibits a tensor product factorization of a corresponding tilting module under the assumption p >= 2h − 2, where h is the Coxeter number. In chapter 4, we remove the restriction on p and consider some variations involving the more general notion of (p,r)−minuscule weights.
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Usher, Andrew Edward Ronald. "Cluster points and cohomology for abelian groups." Thesis, Queen Mary, University of London, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.397960.

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12

Rizkallah, John. "Bounding cohomology for low rank algebraic groups." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267214.

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Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.
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13

Eastridge, Samuel Vance. "First Cohomology of Some Infinitely Generated Groups." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/77517.

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The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case.
Ph. D.
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14

Chen, Yu Qing. "Farrell cohomology of automorphism groups of free groups of finite rank /." The Ohio State University, 1997. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487947908404206.

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15

Green, David John. "On the cohomology of certain finite simple groups." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239184.

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16

Hutchinson, Samuel M. A. "The Morava cohomology of finite general linear groups." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/20464/.

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In this thesis, for all finite heights n and odd primes p, we compute the Morava E-theory and Morava K-theory of general linear groups over finite fields F of order q ≡ 1 mod p. We rephrase the problem in terms of V_*, the graded groupoid of vector spaces over F, and focus on the graded algebra and coalgebra structures induced from the direct sum functor. We use character theory to determine the ranks of E^0BV_* and K^0BV_*, and use this information to reverse engineer the Atiyah-Hirzebruch spectral sequence for K^*BV_* and K_*BV_*. We then use this result in two ways: we deduce that the algebra and coalgebra structures are free commutative and cofree cocommutative respectively, and we identify a lower bound for the nilpotence of the canonical top normalised Chern class in K^0BV_{p^k}. Following this we make use of algebro-geometric and Galois theoretic techniques to determine the indecomposables in Morava E-theory and K-theory, before using this calculation in conjunction with K-local duality and the nilpotence lower bound to determine the primitives of the coalgebra structure in Morava K-theory. Along the way, we show that E^0BV_* and K^0BV_* have structures similar to that of a graded Hopf ring, but with a modified version of the compatibility relation. We call such structures "graded faux Hopf rings".
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17

Haller, Sergei. "Computing Galois cohomology and forms of linear algebraic groups." Giessen Giessener Elektronische Bibliothek, 2005. http://geb.uni-giessen.de/geb/volltexte/2005/2474/index.html.

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18

Lafferty, Matthew J. "Eichler-Shimura cohomology groups and the Iwasawa main conjecture." Thesis, The University of Arizona, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3702136.

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Ohta has given a detailed study of the ordinary part of p-adic Eichler-Shimura cohomology groups (resp., generalized p-adic Eichler-Shimura cohomology groups) from the perspective of p-adic Hodge theory. Assuming various hypotheses, he is able to use the structure of these groups to give a simple proof of the Iwasawa main conjecture over Q. The goal of this thesis is to extend Ohta’s arguments with a view towards removing these hypotheses.

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19

Xu, Mingzhi. "On Cohomology Groups of Global Units in Zdp-Extensions /." The Ohio State University, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487864986611668.

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20

Lafferty, Matthew John. "Eichler-Shimura Cohomology Groups and the Iwasawa Main Conjecture." Diss., The University of Arizona, 2015. http://hdl.handle.net/10150/556816.

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Ohta has given a detailed study of the ordinary part of p-adic Eichler-Shimura cohomology groups (resp., generalized p-adic Eichler-Shimura cohomology groups) from the perspective of p-adic Hodge theory [O₁, O₂, O₃]. Assuming various hypotheses, he is able to use the structure of these groups to give a simple proof of the Iwasawa main conjecture over Q [O₂, O₃, O₄, O₅]. The goal of this thesis is to extend Ohta’s arguments with a view towards removing these hypotheses.
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21

Hamilton, Martin. "Finiteness conditions in group cohomology." Thesis, Connect to e-thesis, 2008. http://theses.gla.ac.uk/182/.

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Thesis (Ph.D.) - University of Glasgow, 2008.
Ph.D. thesis submitted to the Faculty of Information and Mathematical Sciences, University of Glasgow, 2008. Includes bibliographical references. Print version also available.
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22

Edmundo, Mario Jorge. "O-minimal expansions of groups." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312447.

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23

Guillot, Pierre. "Representations and Cohomology of Groups -- Topics in algebra and topology." Habilitation à diriger des recherches, Université de Strasbourg, 2012. http://tel.archives-ouvertes.fr/tel-00732874.

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Mémoire rédigé en vue de l'obtention de l'habilitation à diriger les recherches. Il donne un résumé de mon activité de recherche (anneaux de Chow, classes de Stiefel-Whitney, algèbres de Hopf, entrelacs, K-théorie de Milnor).
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24

Clarke, Nicholas. "Cohomology of groups with coefficients in a non-trivial module." Thesis, University of Manchester, 2016. https://www.research.manchester.ac.uk/portal/en/theses/cohomology-of-groups-with-coefficients-in-a-nontrivial-module(497ed9be-bac2-4b73-9fd1-f4f21bf5b610).html.

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Reeder, Mark Stephen. "The Steinberg module and the top cohomology of arithmetic groups /." The Ohio State University, 1988. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487587604133071.

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26

Griffin, James Thomas. "Automorphisms of free products of groups." Thesis, University of Cambridge, 2013. https://www.repository.cam.ac.uk/handle/1810/244265.

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The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.
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Liedtke, Christian. "On fundamental groups of Galois closures of generic projections." Bonn : Mathematisches Institut der Universität, 2004. http://catalog.hathitrust.org/api/volumes/oclc/62768237.html.

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Cha, Byungchul. "Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves." Available to US Hopkins community, 2003. http://wwwlib.umi.com/dissertations/dlnow/3080634.

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Ozel, Cenap. "On the complex cobordism of flag varieties associated to loop groups." Thesis, University of Glasgow, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.241783.

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Cheng, Jian-Jun. "Restrictions of invariants of reflections and dirac cohomology /." View abstract or full-text, 2004. http://library.ust.hk/cgi/db/thesis.pl?MATH%202004%20CHENG.

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Santacruz, Camilo Andres Angulo. "A cohomology theory for Lie 2-algebras and Lie 2-groups." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-15022019-084657/.

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In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.
Nesta tese, nós introduzimos uma nova teoria de cohomologia associada às 2-álgebras de Lie e uma nova teoria de cohomologia associada aos 2-grupos de Lie. Prova-se que estas teorias de cohomologia estendem as teorias de cohomologia clássicas de álgebras de Lie e grupos de Lie em que os seus segundos grupos classificam extensões. Finalmente, usaremos estos fatos junto com um morfismo de van Est adaptado para encontrar uma nova prova da integrabilidade das 2-álgebras de Lie.
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Callegaro, Filippo. "Cohomology of finite and affine type Artin groups over Abelian representation /." Pisa, Italy : Edizioni della normale, 2009. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=017728632&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.

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Katayama, Shin-ichi. "A Theorem on the Cohomology of Groups and some Arithmetical Applications." 京都大学 (Kyoto University), 1985. http://hdl.handle.net/2433/86360.

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Callegaro, Filippo. "Cohomology of finite and affine type Artin groups over abelian representations." Doctoral thesis, Scuola Normale Superiore, 2007. http://hdl.handle.net/11384/85685.

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Joecken, Kyle. "Dimension of Virtually Cyclic Classifying Spaces for Certain Geometric Groups." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1374085871.

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Grieder, Ralph. "On the symplectic classes in the cohomology of the mapping class groups /." [S.l.] : [s.n.], 1996. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=11677.

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Weber, Patrick. "Cohomology groups on hypercomplex manifolds and Seiberg-Witten equations on Riemannian foliations." Doctoral thesis, Universite Libre de Bruxelles, 2017. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/252914.

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The thesis comprises two parts. In the first part, we investigate various cohomological aspects of hypercomplex manifolds and analyse the existence of special metrics. In the second part, we define Seiberg-Witten equations on the leaf space of manifolds which admit a Riemannian foliation of codimension four.
Doctorat en Sciences
info:eu-repo/semantics/nonPublished
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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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Kumon, Asuka. "On derivatives of L-series, p-adic cohomology and ray class groups." Thesis, King's College London (University of London), 2017. https://kclpure.kcl.ac.uk/portal/en/theses/on-derivatives-of-lseries-padic-cohomology-and-ray-class-groups(1ec486f7-f9aa-45c2-a245-976d4dd9d10f).html.

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We investigate the explicit Galois structure of ray class groups. We also study consequences of the structural results we obtain concerning the validity (or otherwise) of Leopoldt's Conjecture and the existence of families of congruence relations between the values of Dirichlet L-series at z = 1.
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Ando, Matthew. "Operations in complex-oriented cohomology theories related to subgroups of formal groups." Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/13230.

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Melo, Heather Aurora Florence de. "Totally real Galois representations in characteristic 2 and arithmetic cohomology /." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1048.pdf.

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TERRAGNI, TOMMASO. "Hecke algebras associated to coxeter groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2012. http://hdl.handle.net/10281/29634.

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In this thesis, we study cohomological properties of Hecke algebras $H_q(W,S)$ associated with arbitrary Coxeter groups $(W,S)$. Under mild conditions, it is possible to canonically define the Euler characteristic of such an algebra. We define an almost-canonical complex of $H$-modules that allows one to compute the Euler characteristic of $H$. It turns out that the Euler characteristic of the algebra has an interpretation as a combinatorial object attached to the Coxeter group: indeed, for suitable choices of the base ring, it is the inverse of the Poincaré series. Some other results about Coxeter groups are proved, in particular one new characterization of minimal non-spherical, non-affine types is given.
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Sato, Takashi. "The T-equivariant Integral Cohomology Ring of F4/T." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199076.

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44

Deshpande, D. V. "Topological methods in algebraic geometry : cohomology rings, algebraic cobordism and higher Chow groups." Thesis, University of Cambridge, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.598515.

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This thesis is divided into three chapters. The first chapter is about the cohomology ring of the space of rotational functions. In the second chapter, we define algebraic cobordism of classifying spaces, Ω*(BG) and G-equivariant algebraic cobordism Ω*G(-) for a linear algebraic group G. We prove some properties of the coniveau filtration on algebraic cobordism, denoted Fj(Ω*(-)); which are required for the definition to work. We show that G-equivariant cobordism satisfies the localization exact sequence. We compute Ω*(BG) for algebraic groups over the complex numbers corresponding to classical Lie groups GL(n), SL(n), Sp(n), O(n) and SO(2n + 1). We also compute Ω*(BG) when G is a finite abelian group. A finite non-abelian group for which we compute Ω*(BG) is the quaternion group of order 8. In all the above cases we check that Ω*(BG) is isomorphic to MU*(BG). The third chapter is work-in-progress on Steenrod operations on higher Chow groups. Voevodsky has defined motivic Steenrod operations on motivic cohomology and used them in his proof of the Minor Conjecture.
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45

Nave, Lee Stewart. "The cohomology of finite subgroups of Morava stabilizer groups and Smith-Toda complexes /." Thesis, Connect to this title online; UW restricted, 1999. http://hdl.handle.net/1773/5803.

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46

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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47

Fukumoto, Yoshiyasu. "On the Strong Novikov Conjecture of Locally Compact Groups for Low Degree Cohomology Classes." Kyoto University, 2016. http://hdl.handle.net/2433/217729.

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48

Gandhi, Raj. "Oriented Cohomology Rings of the Semisimple Linear Algebraic Groups of Ranks 1 and 2." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42566.

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In this thesis, we compute minimal presentations in terms of generators and relations for the oriented cohomology rings of several semisimple linear algebraic groups of ranks 1 and 2 over algebraically closed fields of characteristic 0. The main tools we use in this thesis are the combinatorics of Coxeter groups and formal group laws, and recent results of Calm\`es, Gille, Petrov, Zainoulline, and Zhong, which relate the oriented cohomology rings of flag varieties and semisimple linear algebraic groups to the dual of the formal affine Demazure algebra.
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49

Lond, Daniel. "On Reductive Subgroups of Algebraic Groups and a Question of Külshammer." Thesis, University of Canterbury. Mathematics and Statistics, 2013. http://hdl.handle.net/10092/8033.

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This Thesis is motivated by two problems, each concerning representations (homomorphisms) of groups into a connected reductive algebraic group G over an algebraically closed field k. The first problem is due to B. Külshammer and is to do with representations of finite groups in G: Let Γ be a finite group and suppose k has characteristic p. Let Γp be a Sylow p-subgroup of Γ and let ρ : Γp → G be a representation. Are there only finitely many conjugacy classes of representations ρ' : Γ → G whose restriction to Γp is conjugate to ρ? The second problem follows the work of M. Liebeck and G. Seitz: describe the representations of connected reductive algebraic H in G. These two problems have been settled as long as the characteristic p is large enough but not much is known in the case where the characteristic p is a so called bad prime for G, which will be the setting for our work. At the intersection of these two problems lies another problem which we call the algebraic version of Külshammer's question where we no longer suppose Γ is finite. This new variation of Külshammer's question is interesting in its own right, and a counterexample may provide insight into Külshammer's original question. Our approach is to convert these problems into problems in the nonabelian 1-cohomology. Let K be a reductive algebraic group, P a parabolic subgroup of G with Levi subgroup L < P, V the unipotent radical of P. Let ρ₀ : K → L be a representation. Then the representations ρ : K → P that equal ρ₀ under the canonical projection P → L are in bijective correspondence with elements of the space of 1-cocycles Z¹(K,V ) where K acts on V by xv = ρ₀(x)vρ₀(x)⁻¹. We can then interpret P- and G-conjugacy classes of representations in terms of the 1-cohomology H¹(K,V ). We state and prove the conditions under which a collection of representations from K to P is a finite union of conjugacy classes in terms of the 1-cohomology in Theorem 4.22. Unlike other approaches, we work directly with the nonabelian 1-cohomology. Even so, we find that the 1-cocycles in Z¹(K,V ) often take values in an abelian subgroup of V (Lemmas 5.10 and 5.11). This is interesting, for the question "is the restriction map of 1-cohomologies H¹(H,V) → H¹(U,V) induced by the inclusion of U in K injective?" is closely linked to the question of Külshammer, and has positive answer if V is abelian and H = SL₂k) (Example 3.2). We show that for G = B4 there is a family of pairwise non-conjugate embeddings of SL₂in G, a direction provided by Stewart who proved the result for G = F4. This is important as an example like this is first needed if one hopes to find a counterexample to the algebraic version of Külshammer's question.
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50

Mierach, Svea Nora Verfasser], and Christoph [Akademischer Betreuer] [Schweigert. "Hochschild cohomology, modular tensor categories and mapping class groups / Svea Nora Mierach ; Betreuer: Christoph Schweigert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2020. http://d-nb.info/1216998140/34.

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