Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

We give a construction of the universal enveloping $$A_\infty $$ A ∞ algebra of a given $$L_\infty $$ L ∞ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\infty $$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

The Cantor–Schröder–Bernstein Theorem for $$\infty $$-groupoids

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or $$\infty $$ ∞ -groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean $$\infty $$ ∞ -topos.

The homotopy types of $$\mathrm{SO}(4)$$-gauge groups

The homotopy types of gauge groups of principal $$\mathrm{SO}(4)$$ SO ( 4 ) -bundles over $$S^{4}$$ S 4 are classified p-locally for every prime p, and partial results are obtained integrally. The method generalizes to deal with any quotient of the form $$(S^{3})^{n}/Z$$ ( S 3 ) n / Z where Z is a subgroup generated by $$(-1,\ldots ,-1)$$ ( - 1 , … , - 1 ) .

The homotopy types of U(n)-gauge groups over lens spaces

We analyse the homotopy types of gauge groups for principal U(n)-bundles over lens spaces and two-dimensional Moore spaces.