Some characterizations of acyclic maps

Raptis, George

doi:10.1007/s40062-019-00231-6

Cited by 12 publications

(14 citation statements)

References 9 publications

(22 reference statements)

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“…The plus construction. Let X be a based connected space in S * (for example, X = BG for a discrete group G) and let X + P denote the plus construction of X associated to a normal perfect subgroup P of π 1 (X, * ) [8,15]. We recall that every connected space is equivalent to BG + P for some discrete group G and a normal perfect subgroup P G by the Kan-Thurston theorem [12].…”

Section: Examplesmentioning

confidence: 99%

Bounded cohomology and homotopy colimits

Raptis¹

2021

Preprint

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Section: Examplesmentioning

confidence: 99%

Bounded cohomology and homotopy colimits

Raptis¹

2021

Preprint

Self Cite

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“…More precisely, X → X + is the initial map that kills the maximal perfect subgroups of the fundamental groups of X, and hence it is an equivalence if and only if the fundamental groups of X are hypoabelian (i.e., have no nontrivial perfect subgroups). We refer to [Rap19] for a discussion of acyclic morphisms in Spc and for a proof of this fact.…”

Section: Quillen's Plus Construction and Group Completionmentioning

confidence: 99%

On the infinite loop spaces of algebraic cobordism and the motivic sphere

Bachmann¹,

Elmanto²,

Hoyois³

et al. 2021

Épijournal De Géométrie Algébrique

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We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic. Comment: 13 pages. v5: published version; v4: final version, to appear in \'Epijournal G\'eom. Alg\'ebrique; v3: minor corrections; v2: added details in the moving lemma over finite fields

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“…We refer to [Rap19] for a review of the main properties of acyclic morphisms. In particular, by [Rap19,Theorem 3.3], the class of acyclic morphisms is closed under colimits and base change, and if X is a space whose fundamental groups have no nontrivial perfect subgroups, then every acyclic morphism X → Y is an equivalence. Lemma 2.1.1.…”

Section: Stable Vector Bundles and K-theorymentioning

confidence: 99%

Modules over algebraic cobordism

Elmanto

Hoyois

Khan

et al. 2020

Forum of Mathematics, Pi

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We prove that the $\infty $ -category of $\mathrm{MGL} $ -modules over any scheme is equivalent to the $\infty $ -category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$ -loop spaces, we deduce that very effective $\mathrm{MGL} $ -modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$ , $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$ .

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Some characterizations of acyclic maps

Cited by 12 publications

References 9 publications

Bounded cohomology and homotopy colimits

Bounded cohomology and homotopy colimits

On the infinite loop spaces of algebraic cobordism and the motivic sphere

Modules over algebraic cobordism

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