Controlled homotopy topological structures

Controlled Homotopy Topological Structures

Let p : E —> B be a locally trivial fiber bundle between closed manifolds where dim E > 5 and B has a handlebody decomposition. A controlled homotopy topological structure (or a controlled structure^ for short) is a map / : M —> E where M is a closed manifold of the same dimension as E and / is a p~ (ε)-equivalence for every ε > 0 (see §2). It is the purpose of this paper to develop an obstruct...

Homotopy Coherent Structures

Naturally occurring diagrams in algebraic topology are commutative up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such “up to homotopy” settings. The firs...

Nanoparticle controlled soft complex structures with topological defects

Homotopy Gerstenhaber Algebras and Topological Field Theory

We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and “string vertices” by obtaining new results on Vassiliev invariants of knots and double...

Homotopy Actions by Topological Actions . Ii

A homotopy action of a group G on a space X is a homomorphism from G to the group HAUT(X) of homotopy classes of homotopy equivalences of X. George Cooke developed an 'obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G and maps between them that are...