Higher homotopy normalities in topological groups

Higher Groups in Homotopy Type Theory

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups...

Homotopy Variation Map and Higher Homotopy Groups of Pencils

We prove in a synthetic manner a general Zariski-van Kampen type theorem for higher homotopy groups of pencils on singular complex spaces.

On Higher Homotopy Groups of Pencils

We consider a large, convenient enough class of pencils on singular complex spaces. By introducing variation maps in homotopy, we prove in a synthetic manner a general Zariski-van Kampen type result for higher homotopy groups, which in homology is known as the “second Lefschetz theorem”.

Higher Homotopy of Groups Definable in O-minimal Structures

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their unive...

Hypersurface complements , Milnor fibers and higher homotopy groups of arrangments