Given a simplicial pair (X, A), complex Y, and map $$f:A \rightarrow Y$$ , does f have an extension to X? We show that for fixed this question is algorithmically decidable all X, A, if Y has the rational homotopy type of H-space. As corollary, many questions related bundle structures over finite are likely decidable. Conversely, other at least as hard certain special cases Hilbert’s tenth probl...

Consider the Fukaya category associated to a Lefschetz fibration. It turns out that the Floer cohomology of the monodromy around∞ gives rise to natural transformations from the Serre functor to the identity functor, in that category. We pay particular attention to the implications of that idea for Lefschetz pencils.

Homotopy type theory is a new branch of mathematics which merges insights from abstract homotopy theory and higher category theory with those of logic and type theory. It allows us to represent a variety of mathematical objects as basic type-theoretic constructions, higher inductive types. We present a proof that in homotopy type theory, the torus is equivalent to the product of two circles. Th...

Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufcient conditions for the proper part of P being homotopy equivalent to the suspension of the proper part of Q. An application of this lemma is a uniied proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result by ZIEGLER) and single step inclusion order (whi...

Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincare conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product F x [0, l] or a half-open product F X [O, 1) where F is a 2-manifold is actually homeomorphic to Fx [0, l] or F x [0, 1), respectively. By defining a concept for noncompact manifolds si...