Algebraic K-Theory and Its Applications

Rosenberg, Jonathan

doi:10.1007/978-1-4612-4314-4

Cited by 339 publications

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“…it is independent of the choice of D. The complex C is homotopy S-finite if and only if its finiteness obstruction is trivial; see [6,Theorem 1.7.12] for a textbook proof. In this sense, the algebraic K-theory detects homotopy finiteness of finitely dominated chain complexes.…”

Section: Finitely Dominated Chain Complexesmentioning

confidence: 99%

“…It follows that C is homotopy equivalent to a summand of the chain complex (9), which is quasi-isomorphic, via (8) and (6), to the finite complex L of R n−1 -modules. Consequently, C, considered as an R n−1 -module complex, is a retract up to homotopy of the chain complex L. Indeed, the complex (9) can be replaced, up to quasiisomorphism, by a bounded below complex of projective R n−1 -modules, which is quasi-isomorphic, and hence chain homotopy equivalent, to L. Using the fact that C is a bounded below complex of projective R n−1 -modules as well it is then standard homological algebra to construct the desired maps of complexes α : C -L and β : L -C together with a chain homotopy βα id.…”

Section: Invocation Of Corollary 23 Gives Us a Quasi-isomorphismmentioning

confidence: 99%

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Finite Domination and Novikov Rings. Iterative Approach

Hüttemann¹,

Quinn²

2012

Glasgow Math. J.

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Abstract. Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R [x, x −1 ]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C ⊗ L R((x)) and C ⊗ L R((x −1 )) are acyclic, as has been proved by Ranicki (A. Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619-632).are rings of the formal Laurent series, also known as Novikov rings. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.2000 Mathematics Subject Classification. Primary 55U15; Secondary 18G35.Finiteness conditions for chain complexes of modules play an important role in both algebra and topology. For example, given a group G, one might ask whether the trivial G-module ‫ޚ‬ admits a resolution by finitely generated projective ‫[ޚ‬G]-modules; existence of such resolutions is relevant for the study of group homology of G, and has applications in the theory of duality groups [1]. For topologists, finite domination of chain complexes is related, among other things, to questions about finiteness of CW complexes, the topology of ends of manifolds and obstructions for the existence of non-singular closed 1-forms [5,7].A chain complex C of R[x, x −1 ]-modules is called finitely dominated if it is homotopy equivalent, as a complex of R-modules, to a bounded complex of finitely generated projective R-modules. Finite domination of C can be characterised in various ways; Brown considered compatibility of the functors M → H * (C; M) and M → H * (C; M) with products and direct limits, respectively [1, Theorem 1], whereas Ranicki showed that C is finitely dominated if and only if the Novikov homology of C is trivial (see [5, Theorem 2], and Theorem 1.2 in this paper).

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Section: Finitely Dominated Chain Complexesmentioning

confidence: 99%

Section: Invocation Of Corollary 23 Gives Us a Quasi-isomorphismmentioning

confidence: 99%

Finite Domination and Novikov Rings. Iterative Approach

Hüttemann¹,

Quinn²

2012

Glasgow Math. J.

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“…We first immediately have H gr 2 (T g ) ∼ = 0, since T g is the universal central extension of M g and the group M g is perfect (see, e.g., [Ros,Corollary 4.1.18]). …”

Section: Dehn Quandle Of Genus ≥mentioning

confidence: 99%

“…Let BM + g,k denote Quillen plus construction of an Eilenberg-MacLane space of M g,k (see, e.g., [Ros,Chapter 5.2] for the definition). Since M g,k is perfect, the space BM + g,k is simply connected.…”

Section: Dehn Quandle Of Genus ≥mentioning

confidence: 99%

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Homotopical interpretation of link invariants from finite quandles

Nosaka

2015

Topology and its Applications

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This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring polynomial and a Z-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.

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Abelianization and fixed point properties of units in integral group rings

Bächle

Janssens

Jespers

et al. 2022

Mathematische Nachrichten

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Let 𝐺 be a finite group and  (ℤ𝐺) the unit group of the integral group ring ℤ𝐺. We prove a unit theorem, namely, a characterization of when  (ℤ𝐺) satisfies Kazhdan's property (T), both in terms of the finite group 𝐺 and in terms of the simple components of the semisimple algebra ℚ𝐺. Furthermore, it is shown that for  (ℤ𝐺), this property is equivalent to the weaker property FAb (i.e., every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property FA, denoted HFA. More precisely, it is described when all subgroups of finite index in  (ℤ𝐺) have both finite abelianization and are not a nontrivial amalgamated product. A crucial step for this is a reduction to arithmetic groups SL 𝑛 (), where  is an order in a finite-dimensional semisimple ℚ-algebra 𝐷, and finite groups 𝐺, which have the so-called cut property. For such groups 𝐺, we describe the simple epimorphic images of ℚ𝐺. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups E 𝑛 (𝐷) of SL 𝑛 (𝐷). These groups are well understood except in the degenerate case of lower rank, that is, for SL 2 () with  an order in a division algebra 𝐷 with a finite number of units. In this setting, we determine Serre's property FA for E 2 () and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its ℤ-rank.

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Algebraic K-Theory and Its Applications

Cited by 339 publications

References 0 publications

Finite Domination and Novikov Rings. Iterative Approach

Finite Domination and Novikov Rings. Iterative Approach

Homotopical interpretation of link invariants from finite quandles

Abelianization and fixed point properties of units in integral group rings

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