This short note introduces a notion of directed homotopy equivalence (or dihomotopy equivalent) and of “directed” topological complexity (which elaborates on the notion that can be found in e.g. [9]) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in [5]). Also, under mild conditions, directe...

Homotopy Type Theory (HoTT) refers to the homotopical interpretation [1] of Martin-Löf’s intensional, constructive type theory (MLTT) [5], together with several new principles motivated by that interpretation. Voevodsky’s Univalent Foundations program [6] is a conception for a new foundation for mathematics, based on HoTT and implemented in a proof assistant like Coq [2]. Among the new principl...

Some new fixed point theorems for approximable maps are obtained in this paper. Homotopy results, via essential maps, are also presented for approximable maps.

The purpose of this paper is to show that ideas and techniques of the homotopy continuation method can be used to find the complete set of eigenpairs of a symmetric matrix. The homotopy defined by Chow, Mallet- Paret and York [I] may be used to solve this problem with 2""-n curves diverging to infinity which for large n causes a great inefficiency. M. Chu 121 introduced a homotopy equation...

Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our first goal is to explain both and show their equivalence. Our second goal is to generalize this result to enriched categories and homotopy weighted limits, s...

Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.

Certain naturally occurring spaces have isomorphic v 1-periodic homotopy groups. To each is associated a mapping telescope whose ordinary homotopy groups equal the v 1-periodic homotopy groups of the space. It is proved that the mapping telescopes of the spaces are homotopy equivalent .

We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko [11] and Varchenko [23] etc. concerning polynomials on C will be generalized to polynomial functions on any toric variety.

Introduction 1 1. Notation 8 2. Finite group action on products of curves 9 3. The fundamental group of (C1 × C2)/G. 12 4. The structure theorem for fundamental groups of quotients of products of curves 14 5. The classification of standard isotrivial fibrations with pg = q = 0 where X = (C1 × C2)/G has rational double points 20 5.1. The singularities of X 21 5.2. The signatures 23 5.3. The poss...

In the first part of this paper we present a formalization in Agda of the James construction in homotopy type theory. We include several fragments of code to show what the Agda code looks like, and we explain several techniques that we used in the formalization. In the second part, we use the James construction to give a constructive proof that π4(S ) is of the form Z/nZ (but we do not compute ...