Waldhausen’s Nil groups and continuously controlled K-theory

Hans Munkholm; Stratos Prassidis

Displaying similar documents to “Waldhausen’s Nil groups and continuously controlled K-theory”

A note on singular homology groups of infinite products of compacta

Kazuhiro Kawamura (2002)

Fundamenta Mathematicae

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Let n be an integer with n ≥ 2 and $X_{i}$ be an infinite collection of (n-1)-connected continua. We compare the homotopy groups of $Σ (\prod_{i} X_{i})$ with those of $\prod_{i} Σ X_{i}$ (Σ denotes the unreduced suspension) via the Freudenthal Suspension Theorem. An application to homology groups of the countable product of the n(≥ 2)-sphere is given.

Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Robert Lipshitz, David Treumann (2016)

Journal of the European Mathematical Society

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Let $A$ be a dg algebra over $𝔽_{2}$ and let $M$ be a dg $A$ -bimodule. We show that under certain technical hypotheses on $A$ , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_{A}^{L} M$ and converges to the Hochschild homology of $M$ . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

Injective comodules and Landweber exact homology theories

Mark Hovey (2007)

Fundamenta Mathematicae

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We classify the indecomposable injective E(n)⁎E(n)-comodules, where E(n) is the Johnson-Wilson homology theory. They are suspensions of the $J_{n, r} = E (n) ⁎ (M_{r} E (r))$ , where 0 ≤ r ≤ n, with the endomorphism ring of $J_{n, r}$ being $\hat{E (r)} * \hat{E (r)}$ , where $\hat{E (r)}$ denotes the completion of E(r).

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...

A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ ¹ (ℤ ₊^{k})$

F. Gourdeau, Z. A. Lykova, M. C. White (2005)

Studia Mathematica

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We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe...

A problem of Kollár and Larsen on finite linear groups and crepant resolutions

Robert Guralnick, Pham Tiep (2012)

Journal of the European Mathematical Society

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The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $\leq 1$ . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation....

Homological computations in the universal Steenrod algebra

A. Ciampella, L. A. Lomonaco (2004)

Fundamenta Mathematicae

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We study the (bigraded) homology of the universal Steenrod algebra Q over the prime field ₂, and we compute the groups $H_{s, s} (Q)$ , s ≥ 0, using some ideas and techniques of Koszul algebras developed by S. Priddy in [5], although we presently do not know whether or not Q is a Koszul algebra. We also provide an explicit formula for the coalgebra structure of the diagonal homology $D ⁎ (Q) = ⨁_{s \geq 0} H_{s, s} (Q)$ and show that D⁎(Q) is isomorphic to the coalgebra of invariants Γ introduced by W. Singer in [6].

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska (2001)

Fundamenta Mathematicae

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Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $(B_{p} (G)) / G$ of the classifying space of the category associated to the G-poset $_{p} (G)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{E ₙ} / (N E ₀ \cap . . . \cap N E ₙ)$ defined over the poset $(s d_{p} (G)) / G$ , where sd is the barycentric subdivision....

L²-homology and reciprocity for right-angled Coxeter groups

Boris Okun, Richard Scott (2011)

Fundamenta Mathematicae

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Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra $_{μ}$ containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define $L ²_{μ}$ -Betti numbers and an $L ²_{μ}$ -Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version...

Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

Plane and layer $(p 2, 2^{l})$ symmetry groups

S. V. Jablan (1990)

Matematički Vesnik

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Homology of origamis with symmetries

Carlos Matheus, Jean-Christophe Yoccoz, David Zmiaikou (2014)

Annales de l’institut Fourier

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Given an origami (square-tiled surface) $M$ with automorphism group $Γ$ , we compute the decomposition of the first homology group of $M$ into isotypic $Γ$ -submodules. Through the action of the affine group of $M$ on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.

Topology of Fatou components for endomorphisms of $ℂ ℙ^{k}$ : linking with the Green’s current

Suzanne Lynch Hruska, Roland K. W. Roeder (2010)

Fundamenta Mathematicae

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Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f : ℂ ℙ^{k} \to ℂ ℙ^{k}$ , when k >1. Classical theory describes U(f) as the complement in $ℂ ℙ^{k}$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ ℙ^{k}$ , we give a definition of linking number between closed loops in $ℂ ℙ^{k} ∖ s u p p S$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H ₁ (ℂ ℙ^{k} ∖ s u p p S)$ . As an application, we use...

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type $𝐀_{1}$

Bo Hou, Yanhong Guo (2015)

Czechoslovak Mathematical Journal

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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let $Λ_{t}$ be the Yoneda algebra of a reconstruction algebra of type $𝐀_{1}$ over a field $. I n t h i s p a p e r, a m i n i m a l p r o j e c t i v e b i m o d u l e r e s o l u t i o n o f$ t $i s c o n s t r u c t e d, a n d t h e$ -dimensions of all Hochschild homology and cohomology groups of $Λ_{t}$ are calculated explicitly.

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

Groups of simple and multiple antisymmetry of similitude in $E^{2}$

S. V. Jablan (1986)

Matematički Vesnik

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Presentations of finite simple groups: a computational approach

Robert Guralnick, William M. Kantor, Martin Kassabov, Alexander Lubotzky (2011)

Journal of the European Mathematical Society

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All finite simple groups of Lie type of rank $n$ over a field of size $q$ , with the possible exception of the Ree groups $^{2} G_{2} (q)$ , have presentations with at most 49 relations and bit-length $O (𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q)$ . Moreover, $A_{n}$ and $S_{n}$ have presentations with 3 generators; 7 relations and bit-length $O (𝚕𝚘𝚐 n)$ , while $𝚂𝙻 (n, q)$ has a presentation with 6 generators, 25 relations and bit-length $O (𝚕𝚘𝚐 n + 𝚕𝚘𝚐 q)$ .

On absolutely-nilpotent of class $k$ groups

Patrizia Longobardi, Trueman MacHenry, Mercede Maj, James Wiegold (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A group $G$ in a variety $V$ is said to be absolutely- $V$ , and we write $G \in A V$ , if central extensions by $G$ are again in $V$ . Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class $A N_{k}$ of absolutely-nilpotent of class $k$ groups. We prove some closure properties of the class $A N_{k}$ and we show that every nilpotent of class $k$ group can be embedded in an $A N_{k}$ -gvoup. We describe all metacyclic $A N_{k}$ -groups and we characterize $2$ -generator and infinite $3$ -generator $A N_{2}$ -groups....

Epics in the category of $T_{2}$ k-groups need not have dense range

W. F. Lamartin (1976)

Colloquium Mathematicae

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