Coarse homology theories and finite decomposition complexity

Bunke, Ulrich; Engel, Alexander; Kasprowski, Daniel; Winges, Christoph

doi:10.2140/agt.2019.19.3033

Cited by 5 publications

(10 citation statements)

References 7 publications

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“…Note that a bounded covering is not a morphism of bornological coarse spaces in general, since it may not be proper. The composition of two bounded coverings is again a bounded covering; see [12, Lemma 2.18]. Remark Conditions (iii) and (v) in Definition 5.1 together are equivalent to the following single condition: for every

B

B (X)

there exists a finite coarsely disjoint partition

{false(B_{α} false)}_{α \in A}

B

, that is, a finite partition

{false(B_{α} false)}_{α \in A}

B

such that

false[B_{α} false] \cap false[B_{α^{'}} false] = \emptyset

for all

α \neq α^{'}

, such that

f_{[B_{α}]} : false[B_{α} false] \to false[f (B_{α}) false]

is an isomorphism of the underlying coarse spaces.…”

Section: A Descent Resultsmentioning

confidence: 99%

“…(4) In Theorem 10.11, we use the geometric assumptions on the subgroups H in order to deduce from [12] that the forget-control maps in the H-equivariant context are equivalences.…”

Section: Inductionmentioning

confidence: 99%

“… (1)Proposition 10.13 reduces the proof to the verification that a certain morphism

L false(S false) \to {boldM}_{A} false(S false)

is an equivalence for every

S

G_{scriptF} Orb

. (2)Proposition 10.14 identifies this morphism with the composition of a descent morphism and a forget‐control map depending on subgroups

H

in the family

scriptF

. (3)In Theorem 10.9, we use the descent result to show that the descent morphism is an equivalence, and therefore reduce the problem to the verification that forget‐control maps are equivalences for subgroups

H

in the family

scriptF

. This step employs transfers. (4)In Theorem 10.11, we use the geometric assumptions on the subgroups

H

in order to deduce from [12] that the forget‐control maps in the

H

‐equivariant context are equivalences. …”

Section: The Main Theoremmentioning

confidence: 99%

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Injectivity results for coarse homology theories

Bunke

Engel

Kasprowski

et al. 2020

Proc. London Math. Soc.

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B

B (X)

there exists a finite coarsely disjoint partition

{false(B_{α} false)}_{α \in A}

B

, that is, a finite partition

{false(B_{α} false)}_{α \in A}

B

such that

false[B_{α} false] \cap false[B_{α^{'}} false] = \emptyset

for all

α \neq α^{'}

, such that

f_{[B_{α}]} : false[B_{α} false] \to false[f (B_{α}) false]

is an isomorphism of the underlying coarse spaces.…”

Section: A Descent Resultsmentioning

confidence: 99%

“…(4) In Theorem 10.11, we use the geometric assumptions on the subgroups H in order to deduce from [12] that the forget-control maps in the H-equivariant context are equivalences.…”

Section: Inductionmentioning

confidence: 99%

“… (1)Proposition 10.13 reduces the proof to the verification that a certain morphism

L false(S false) \to {boldM}_{A} false(S false)

is an equivalence for every

S

G_{scriptF} Orb

. (2)Proposition 10.14 identifies this morphism with the composition of a descent morphism and a forget‐control map depending on subgroups

H

in the family

scriptF

H

in the family

scriptF

. This step employs transfers. (4)In Theorem 10.11, we use the geometric assumptions on the subgroups

H

in order to deduce from [12] that the forget‐control maps in the

H

‐equivariant context are equivalences. …”

Section: The Main Theoremmentioning

confidence: 99%

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Injectivity results for coarse homology theories

Bunke

Engel

Kasprowski

et al. 2020

Proc. London Math. Soc.

Self Cite

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“…We combine this with the results of [BEKW17] and the technical results of the present paper to deduce split injectivity results for the original Farrell-Jones assembly map.…”

mentioning

confidence: 91%

“…1. In [BEKW17] we show that a certain large scale geometric condition called finite decomposition complexity implies that a motivic version of the forget-control map is an equivalence.…”

Section: Introduction 3 Introductionmentioning

confidence: 99%

Equivariant coarse homotopy theory and coarse algebraic K-homology

Bunke¹,

Engel²,

Kasprowski³

et al. 2020

𝐾-Theory in Algebra, Analysis And Topology

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We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K-homology.Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.

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