Shape theory is an extension of homotopy theory which uses the idea of homotopy in its conception. By comparison, the theory of n-shape, which heretofore only has been defined for metrizable compacta, has as its basic notion that of n-homotopy instead of homotopy. We shall demonstrate that the theory of n-shape extends to the class of all Hausdorff compacta.

Given some type of fibration on a 4-manifold X with a torus regular fiber T , we may produce a new 4-manifold XT by performing torus surgery on T . There is a natural way to extend the fibration to XT , but a multiple fiber (nongeneric) singularity is introduced. We construct explicit generic fibrations (with only indefinite fold singularities) in a neighborhood of this multiple fiber. As an ap...

For spaces localized at 2, the classical EHP fibrations [1, 13] ΩS S ΩS ΩS play a crucial role for the computations of the homotopy groups of the spheres [16, 25]. The EHP-fibrations for (p− 1)-cell complexes for p > 2 are given in this article. These fibrations can be regarded as the odd prime analogue of the classical EHP-fibrations by considering the spheres as 1-cell complexes for p = 2. So...

Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated p...

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky...

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformations and solutions of the corresponding Maurer-Cartan equation. In Section 6 we generalize the Maurer-Cartan equation to strongly homotopy Li...

A smooth, projective surface S is called a standard isotrivial fibration if there exists a finite group G which acts faithfully on two smooth projective curves C1 and C2, so that S is isomorphic to the minimal desingularization of T := (C1 × C2)/G, where G acts diagonally on the product. When the action of G is free, then S = T is called a quasi-bundle. In this paper we analyse several numerica...

Floer cohomology groups are usually defined over a field of formal functions (a Novikov field). Under certain assumptions, one can equip them with connections, which means operations of differentiation with respect to the Novikov variable. This allows one to write differential equations for Floer cohomology classes. Here, we apply that idea to symplectic cohomology groups associated to Lefschet...

(1.1) H = {(x, y) ∈ (C) × C : y1y2 + p(x) = z}, Here, p : (C) → C is the superpotential mirror to Y (following [7] or [9]), and z is any regular value of p. H is an affine threefold with trivial canonical bundle. Hence, it has a Fukaya category Fuk(H), whose objects are compact exact Lagrangian submanifolds equipped with gradings and Spin structures. This is an A∞-category over C. Consider the ...