The “fundamental theorem” for the algebraic 𝐾-theory of spaces. III. The nil-term

Klein, John R.; Williams, E. Bruce

doi:10.1090/s0002-9939-08-09293-9

Cited by 3 publications

(8 citation statements)

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“…We follow the notation of [11,12,15]. Recall that a Waldhausen category is a category with cofibrations and weak equivalences satisfying axioms of [24, 1.2].…”

Section: K-theorymentioning

confidence: 99%

“…Similarly as in the linear case, the nil-terms can be identified with a subgroup of the K-theory of a certain nil-category. Here, we follow [15]. As before, let M be the geometric realization of a simplicial monoid M • .…”

Section: Nil-termsmentioning

confidence: 99%

“…induces a map in K-theory whose homotopy fiber is denoted Nil fd ( * , M). The main theorem of [15] is as follows.…”

Section: Nil-termsmentioning

confidence: 99%

“…Proof of Theorem 4.1. Let us briefly recall how the terms NA fd + and Nil fd are identified in [15] (an analogous proof works for NA fd -). First, it is proved that the inclusion functors induce a homotopy fibration sequence [15, Proposition 3.1]:…”

Section: Identifying Verschiebung and Frobenius Operationsmentioning

confidence: 99%

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Operations on the A-theoretic nil-terms

Grunewald

Klein

Macko

2007

Journal of Topology

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For a space X, we define Frobenius and Verschiebung operations on the nil-terms NA fd ± (X) in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we show that the homotopy groups of NA fd ± (X) are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung operation defines a Z[N×]-module structure on the homotopy groups of NA fd ± (X), with N× the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms NA fd ± ( * ) after p-completion for an odd prime p and their homotopy groups as Zp [N×]-modules up to dimension 4p−7. We obtain non-trivial groups only in dimension 2p − 2, where it is finitely generated as a Zp [N×]-module, and in dimension 2p − 1, where it is not finitely generated as a Zp [N×]-module.where A fd (X) denotes the finitely-dominated version of the algebraic K-theory of the space X, BA fd (X) denotes a certain canonical non-connective delooping of A fd (X), and NA fd + (X), and NA fd -(X) are two homeomorphic nil-terms. Thus, the study of A fd (X × S 1 ) splits naturally into studying A fd (X) and the nil-terms. Over time, there has been steady progress in understanding A fd (X) for some X, mostly for X = * (see [14,18,19]), but not much is known about the nil-terms. These are the subjects of the present paper.Recall that the splitting (1.1) is analogous to the splitting of the fundamental theorem of algebraic K-theory of rings [9] for any ring R( 1.2) where K(R) denotes the (−1)-connective K-theory space of the ring R and BK(R) is a certain canonical non-connective delooping of K(R). In fact, for any space X there is a linearization map l : A fd (X) → K(Z[π 1 X]) and the two splittings are natural with respect to this map. We pursue the analogy between the algebraic K-theory of spaces (the non-linear situation) and the algebraic K-theory of rings (the linear situation) further. We define the Frobenius and Verschiebung operations on the nil-terms NA fd ± (X), which are analogs of such operations defined in the linear case. As a consequence, we obtain the following result (compare with [5]).Theorem 1.1. The homotopy groups π * NA fd ± (X) and all of their p-primary subgroups are either trivial or not finitely generated as abelian groups.

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“…We follow the notation of [11,12,15]. Recall that a Waldhausen category is a category with cofibrations and weak equivalences satisfying axioms of [24, 1.2].…”

Section: K-theorymentioning

confidence: 99%

Section: Nil-termsmentioning

confidence: 99%

“…induces a map in K-theory whose homotopy fiber is denoted Nil fd ( * , M). The main theorem of [15] is as follows.…”

Section: Nil-termsmentioning

confidence: 99%

Section: Identifying Verschiebung and Frobenius Operationsmentioning

confidence: 99%

See 2 more Smart Citations

Operations on the A-theoretic nil-terms

Grunewald

Klein

Macko

2007

Journal of Topology

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show abstract

“…The connective K-theory of generalized Laurent extensions of rings is treated in Waldhausen [25,26]. Hüttemann-Klein-Vogell-Waldhausen-Williams [9] proved a Bass-Heller-Swan decomposition for connective algebraic K-theory of spaces on the spectrum level; Klein-Williams [10] identified the relative terms with the K-theory spectrum of homotopy-nilpotent endomorphisms.…”

Section: Introductionmentioning

confidence: 99%