Generalized plus-constructions and fundamental groups

“…If X + is simply connected, then the canonical map X +h → X h is a homotopy equivalence; see [9, 1.7]. Now X → X +h agrees with X → X +HR for a suitable ring R of the form Z P or ⊕ p∈P Z/p, where P is a set of primes; see [1, 1.1] and compare [12,Sect. 4].…”

Section: Propositionmentioning

confidence: 99%

The plus construction, Postnikov towers and universal central module extensions

Blanc

Peschke

2002

Isr. J. Math.

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Given a connected space X, we consider the effect of Quillen's plus construction on the homotopy groups of X in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequence AX → X → X + , we explain the contribution of π n X to π n X + , π n+1 X + and π n AX, π n+1 AX explicitly in terms of the low dimensional homology of π n X regarded as a module over π 1 X. Key ingredients developed here for this purpose are universal Π-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups. 1 IntroductionQuillen's plus construction (cf.[10]), applied to a space X, yields a universal map η : X → X + , which is characterized by the fact that it quotients out the maximal perfect subgroup of π 1 X and induces isomorphisms in all homology theories (including homology with twisted coefficients). In general, a map between connected spaces satisfies this homological condition if and only if its homotopy fiber is acyclic; see [11] and compare [5]. We denote the homotopy fiber of η : X → X + by AX.

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“…Remark 3.5. The modality of Theorem 3.3 can be regarded as an instance of the construction in [2, Example 3.5.3] for the nullification that defines the plus construction (see[11,3]). We emphasize that the modality of Theorem 3.3 is different from the factorization system that arises from the plus construction as a localization in the ∞-category of spaces and whose left class is the class of maps which become equivalences after plus construction.…”

mentioning

confidence: 99%

Some characterizations of acyclic maps

Raptis

2019

J. Homotopy Relat. Struct.

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We discuss two categorical characterizations of the class of acyclic maps between (path-connected) spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined by the class of acyclic maps, and discuss the content of the generalized Blakers-Massey theorem for this modality.

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Generalized plus-constructions and fundamental groups

Cited by 7 publications

References 8 publications

Central extensions and generalized plus-constructions

Central extensions and generalized plus-constructions

The plus construction, Postnikov towers and universal central module extensions

Some characterizations of acyclic maps

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