Stochastic Algebraic de Rham Complexes

The cohomology of groups

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

Every principal G-bundle over X is classified up to equivalence by a homotopy class X -> BG, where BG is the classifying space of G. On the other hand, for every nice topological space X Milnor constructed a strict model of its loop space (Omega) over tildeX, that is a group. Moreover, the morphisms of topological groups (Omega) over tildeX -> G generate all the G-bundles over X up to equivalence. In this paper, we show that the relation between Milnor's loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological K-theory, group cohomology, and homotopy theory.

On the cohomology of Coxeter groups and their finite parabolic subgroups. II by T. Akita Linear source modules and trivial source modules by R. Boltje Resolutions de foncteurs de Mackey by S. Bouc Homotopy uniqueness of $BPU(3)$ by C. Broto and A. Viruel Lowering operators for $GL(n)$ and quantum $GL(n)$ by J. Brundan Homomorphisms in higher complexity quotient categories by J. F. Carlson and W. W. Wheeler On the homotopy theory of $p$-completed classifying spaces by F. R. Cohen and R. Levi $H^*V$ is of bounded type over $\mathcal {A} (p)$ by M. D. Crossley Classifying spaces of central group extensions by J. Dietz and T. Ratliff Sharp homology decompositions for classifying spaces of finite groups by W. G. Dwyer On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$ by P. Fong and R. J. Milgram Transfer and Chern classes for extraspecial $p$-groups by D. J. Green and P. A. Minh The infinite order of the symplectic classes in the cohomology of the stable mapping class group by R. Grieder A variant of the proof of the Landweber Stong conjecture by H.-W. Henn Unstable modules over the Steenrod algebra and cohomology of groups by H.-W. Henn The category of Mackey functors for a compact Lie group by L. G. Lewis, Jr. Comparison of extensions of modules for algebraic groups and quantum groups by Z. Lin The defect of the completion of a Green functor by F. Luca Simple modules of Ariki-Koike algebras by A. Mathas On the geometry and cohomology of the simple groups $G_2(q)$ and $^3D_4(q)$: II by R. J. Milgram On the relation between simple groups and homotopy theory by R. J. Milgram Varieties for $G_rT$-modules by D. K. Nakano On defect groups of the Mackey algebras for finite groups by F. Oda Applications of stable classifying space theory to group cohomology by S. Priddy The spectra $k$ and $kO$ are not Thom spectra by Yu. B. Rudyak The code of a regular generalized quadrangle of even order by N. S. N. Sastry and P. Sin Quivers of blocks with normal defect group by M. Schaps, D. Shapira, and O. Shlomo Cohomology of a homogeneous space $E_6/T^1\cdot SU(6)$ by T. Watanabe Problem session, Seattle 1996 by D. Benson.

In this paper we consider non-abelian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -cohomology for groups with coefficients in other groups. We prove versions of the ‘five lemma’ arising from this situation. We go on to show that a connected unipotent algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acted on morphically by a connected algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admits a filtration with successive quotients having the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. First, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a connected, reductive algebraic group with Borel subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a unipotent algebraic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -group, we show the restriction map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis upper G comma upper Q right-parenthesis right-arrow upper H Superscript 1 Baseline left-parenthesis upper B comma upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1(G,Q)\to H^1(B,Q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an isomorphism. We also show that this situation admits a notion of rational stability and generic cohomology. We use these results to obtain corollaries about complete reducibility and subgroup structure.

This article delves into algebraic topology, specifically (co)homology theory, which is essential in various mathematical fields. It explores different types of (co)homology groups such as Hochschild, cyclic, reflexive, and dihedral, focusing on dihedral cohomology generated by the dihedral group operating on simplicial complexes. The text discusses when dihedral and reflexive cohomology vanishes and provides examples, along with proving a homomorphism between dihedral cohomology groups. Mathematical concepts like Banach spaces, C*-algebras, and tensor products are elucidated to understand the algebraic structures involved. Theorems and proofs establish the relationships and properties of these cohomology groups, culminating in a canonical isomorphism theorem for the product of two algebras. Overall, the article aims to provide a comprehensive exploration of dihedral cohomology and its interplay with other algebraic structures, offering insights into its applications and theoretical foundations. Additionally, it studies the (co)homology theory of C*-algebras, focusing on the triviality of cohomology groups of operator algebras and discovering the canonical isomorphism between any two unital K -algebras A and A′: HDn(A×A′)≅HDn(A)⊕HDn(A′).

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Zm which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of Zm. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m = 2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.

We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky’s h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.

We establish a connection between continuous K-theory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous K-groups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly non-vanishing continuous K-group and the highest possibly non-vanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible Zariski-Riemann space with respect to different topologies; namely, the rh-topology which is related to K-theory as well as the Zariski topology whereon the cohomology groups in question rely.

On cohomology of crystallographic groups with cyclic holonomy of split type

For any holomorphic function f\colon X\to\mathbb{C} on a complex manifold X , we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on X . These will be sheaves on the real oriented blow-up space of X along f . We show that, in the context of the irregular Riemann–Hilbert correspondence of D’Agnolo–Kashiwara, these objects recover the classical de Rham complexes with moderate growth and rapid decay associated to a holonomic \mathcal{D}_{X} -module. In order to prove the latter, we resolve a recent conjectural duality of Sabbah between these de Rham complexes of holonomic \mathcal{D}_{X} -modules with growth conditions along a normal crossing divisor by making the connection with a classic duality result of Kashiwara–Schapira between certain topological vector spaces. Via a standard dévissage argument, we then prove Sabbah’s conjecture for arbitrary divisors. As a corollary, we recover the well-known perfect pairing between the algebraic de Rham cohomology and rapid decay homology associated to integrable connections on smooth varieties due to Bloch–Esnault and Hien.

We put the theory of interacting topological crystalline phases on a systematic footing. These are topological phases protected by space-group symmetries. Our central tool is an elucidation of what it means to "gauge" such symmetries. We introduce the notion of a "crystalline topological liquid", and argue that most (and perhaps all) phases of interest are likely to satisfy this criterion. We prove a Crystalline Equivalence Principle, which states that in Euclidean space, crystalline topological liquids with symmetry group $G$ are in one-to-one correspondence with topological phases protected by the same symmetry $G$, but acting *internally*, where if an element of $G$ is orientation-reversing, it is realized as an anti-unitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological (SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants which may contribute phases beyond group cohomology, and so we conjecture our classification is complete.

We define and study (derived) character maps of finite-dimensional representations of 1-groups.As models for 1-groups we take homotopy simplicial groups, ie the homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch ( 2002)).We introduce cyclic, symmetric and representation homology for "group algebras" kOE of such groups and construct canonical trace maps (natural transformations) relating these homology theories.We show that, in the case of one-dimensional representations, our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces known in stable homotopy theory.Using this topological interpretation, we deduce some algebraic results on representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring k is a field of characteristic zero.We also study the stable behavior of the derived character maps of n-dimensional representations as n ! 1, in which case we show that these maps "converge" to become isomorphisms.18A25, 18G15, 19D55, 55N35; 14A30, 55P42

In this first chapter, we introduce the basic objects: algebraic sets in affine space, the corresponding affine algebras, and algebraic groups. We proceed in a completely elementary way and assume only some very basic knowledge about groups, polynomial rings, quotient rings, and fields. The general theory will be mainly illustrated with examples concerning hypersurfaces and algebraic groups.

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.