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Jerome Scherer

Publications and source records attributed to Jerome Scherer.

At least 19 recordsLinked to original sources
arXiv2023

Relative plus constructions

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X, H)$ consisting of a connected space $X$ and an $h$-perfect normal subgroup $H$ of the fundamental group $π_1(X)$ an $h$-acyclic map $X \rightarrow X^{+h}_H$ inducing the quotient by $H$ on the fundamental group. When $h$ is an ordinary homology theory with coefficients in a commutative ring with unit $R$, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen's plus construction. We also clarify the necessity to use a strongly $R$-perfect group $H$ in characteristic zero.

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arXiv2020

Conjugation Spaces are Cohomologically Pure

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.

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arXiv2017

Cellular covers of local groups

We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.

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arXiv2017

Relative homological algebra via truncations

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class I of "injective objects". We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A; I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.

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arXiv2016

Homotopy excision and cellularity

Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy pull-back. This difference is measured in terms of the homotopy fibers of the original maps. Restricting our attention to the connectivity of these maps, we recover the classical Blakers-Massey Theorem.

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arXiv2015

Localization genus

Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.

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arXiv2014

Homotopy colimits of nilpotent spaces

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z_k X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections, and that any nilpotent space for which the space of pointed self-maps map_*(X,X) is "canonically" discrete must be aspherical.

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arXiv2013

Conditionally flat functors on spaces and groups

Consider an extension of groups 1 -> K -> G -> Q -> 1 which enjoys the property that the quotient by the lower central series Gamma_{c+1} produces another extension 1 -> K/ Gamma_{c+1} K -> G /Gamma_{c+1} G -> Q / Gamma_{c+1} Q -> 1, of nilpotent groups of class c. We say that the extension is Gamma_{c+1}-flat. Let us pull back the original extension along any homomorphism X -> Q. Does the pullback extension enjoy the same Gamma_{c+1}-flatness property? To answer this question we consider not only quotients by the lower central series, but any localization functor in the category of groups. In fact we start by studying the analogous question for spaces, where we replace extensions by fibration sequences. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. In the category of groups, nullifications also enjoy this property, and so do all epireflections arising from a variety of groups. In particular the answer to the question about the nilpotent quotients is positive.

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arXiv2012

Spaces with Noetherian cohomology

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

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arXiv2011

Goodwillie calculus and Whitehead products

We prove that iterated Whitehead products of length (n+1) vanish in any value of an n-excisive functor in the sense of Goodwillie. We compare then different notions of homotopy nilpotency, from the Berstein-Ganea definition to the Biedermann-Dwyer one. The latter is strongly related to Goodwillie calculus and we analyze the vanishing of iterated Whitehead products in such objects.

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arXiv2009

Can one classify finite Postnikov pieces?

We compare the classical approach of constructing finite Postnikov systems by k-invariants and the global approach of Dwyer, Kan, and Smith. We concentrate on the case of 3-stage Postnikov pieces and provide examples where a classification is feasible. In general though the computational difficulty of the global approach is equivalent to that of the classical one.

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arXiv2009

Noetherian loop spaces

The class of loop spaces whose mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as the infinite complex projective space and 3-connected covers of compact Lie groups. We study the cohomology of the classifying space BX of such an object and prove it is as small as expected, that is, comparable to that of BCP^\infty. We also show that BX differs basically from the classifying space of a p-compact group in a single homotopy group. This applies in particular to 4-connected covers of classifying spaces of Lie groups and sheds new light on how the cohomology of such an object looks like.

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arXiv2008

Homotopy exponents for large H-spaces

We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.

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arXiv2007

A connection between cellularization for groups and spaces via two-complexes

Let $M$ denote a two-dimensional Moore space (so $H_2(M; \Z) = 0$), with fundamental group $G$. The $M$-cellular spaces are those one can build from $M$ by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The question we address here is to characterize the class of $M$-cellular spaces by means of algebraic properties derived from the group $G$. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.

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arXiv2006

Homology exponents for H-spaces

We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of the integral homology. Our main result states if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2^r, S^1, K(Z, 2), and K(Z,3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the same methods that simply connected $H$-spaces whose mod 2 cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod 2 finite H-spaces with copies of K(Z, 2) and K(Z,3).

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arXiv2006

Generalized orientations and the Bloch invariant

For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K_3(C) Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We also discuss the non-compact case, in which there appears a Z/2-ambiguity.

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arXiv2006

Deconstructing Hopf spaces

We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.

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