Covering groups of non-connected topological groups revisited

R. Brown; O. Mucuk

Home/Archive/arXiv

arXiv · math/0009021

Covering groups of non-connected topological groups revisited

R. Brown·O. Mucuk

View original source ↗Read source document ↗Export metadata ↓

Abstract

In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this result and relate it to the theory of obstructions to group extensions. The methods use: the equivalence between covering maps of X and covering groupoids of the fundamental groupoid of X; the equivalence between group groupoids and crossed modules; and descriptions of cohomology in terms of crossed complexes.

Explore related subjects

math.AT math.CT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

R. Brown, O. Mucuk. 2006-12-18. Covering groups of non-connected topological groups revisited. https://arxiv.org/abs/math/0009021

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The symmetric hit problem in four variables: Relative cohits and stabilizer-fixed preimages

In this work, we study the symmetric hit problem in four variables by analyzing the relative cohit space, defined as the kernel of the inclusion-induced map from symmetric to ordinary cohits. We establish that this relative space is isomorphic to a kernel in the first group cohomology of the permutation group and, equivalently, to an $\operatorname{Ext}^1$ kernel for the general linear group, equipped with a canonical action of the normalizer quotient. We use explicit Lucas--Cartan identities to construct unconditional fixed Steenrod preimages for nontrivial stabilizers, resolving infinite families of monomials strictly above the minimal-spike weight. Using the ordinary and symmetric Kameko isomorphisms, we reduce the determination of complete bases in two generic degree families to exact calculations in degrees one, three, six, ten, and sixteen. The relative space first becomes nonzero in degree fifteen, where it is one-dimensional, and it is an indecomposable two-dimensional normalizer module in degree eighteen. In the family $5\cdot2^t-4$, the symmetric cohit dimension stabilizes at six and the relative dimension at one for $t\geq2$; in the family $7\cdot2^t-4$, they are four and zero for $t\geq1$. We give complete bases by explicit exponent formulas and prove Janfada's monomial implication [7, Conjecture 1] in every degree of both families. We extend the known optimal bound for ordinary Kameko isomorphisms to every permutation-invariant cohit space, prove its optimality there, and transport the entire symmetric hit basis by a direct-sum formula.

math.AT↗

arXiv2026

Postnikov invariants for rational spaces

In this paper we construct a theory of Postnikov invariants for (complete) homotopy Lie algebras that is preserved by the integration functor of Robert-Nicoud and Vallette. This result is in the same vein as Berglund's theorem, which relates the homology groups underlying a nilpotent homotopy Lie algebra to the homotopy groups of the associated space. We then deduce an integration functor from filtered Chevalley-Eilenberg cohomology groups to group cohomology groups.

math.AT↗

arXiv2026

A short proof of the Segal conjecture for $C_p$

We give a new proof of the generalized Segal conjecture for the cyclic group of prime order p, i.e. the fact that the Tate diagonal of any bounded below spectrum is a p-completion. The idea, inspired by Hahn and Wilson's argument, is to use descent along the equivariant map from the p-fold tensor power of the field with p elements to its topological Hochschild homology. The only computational input that we use is the calculation of the homotopy rings of this topological Hochschild homology and its Tate construction, together with the map induced by the cyclotomic Frobenius.

math.AT↗