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Maps into the torus and minimal coincidence sets for homotopies

D. L. Goncalves, M. R. Kelly (2002)

Fundamenta Mathematicae

Let X,Y be manifolds of the same dimension. Given continuous mappings f i , g i : X → Y , i = 0,1, we consider the 1-parameter coincidence problem of finding homotopies f t , g t , 0 ≤ t ≤ 1, such that the number of coincidence points for the pair f t , g t is independent of t. When Y is the torus and f₀,g₀ are coincidence free we produce coincidence free pairs f₁,g₁ such that no homotopy joining them is coincidence free at each level. When X is also the torus we characterize the solution of the problem in terms of the Lefschetz...

Mathematical Models of Abstract Systems: Knowing abstract geometric forms

Jean-Pierre Marquis (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Scientists use models to know the world. It is usually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models...

Matrix factorizations and singularity categories for stacks

Alexander Polishchuk, Arkady Vaintrob (2011)

Annales de l’institut Fourier

We study matrix factorizations of a potential W which is a section of a line bundle on an algebraic stack. We relate the corresponding derived category (the category of D-branes of type B in the Landau-Ginzburg model with potential W) with the singularity category of the zero locus of W generalizing a theorem of Orlov. We use this result to construct push-forward functors for matrix factorizations with relatively proper support.

Matrix problems and stable homotopy types of polyhedra

Yuriy Drozd (2004)

Open Mathematics

This is a survey of the results on stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author [3, 5, 6]. The proofs are based on the technique of matrix problems (bimodule categories).

Maximal equicontinuous factors and cohomology for tiling spaces

Marcy Barge, Johannes Kellendonk, Scott Schmieding (2012)

Fundamenta Mathematicae

We study the homomorphism induced on cohomology by the maximal equicontinuous factor map of a tiling space. We will see that in degree one this map is injective and has torsion free cokernel. We show by example, however, that, in degree one, the cohomology of the maximal equicontinuous factor may not be a direct summand of the tiling cohomology.

Measurable cardinals and fundamental groups of compact spaces

Adam Przeździecki (2006)

Fundamenta Mathematicae

We prove that all groups can be realized as fundamental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group G is nonmeasurable then the compact space K such that G = π₁K may be chosen so that it is path connected.

Miller spaces and spherical resolvability of finite complexes

Jeffrey Strom (2003)

Fundamenta Mathematicae

Let 𝒜 be a fixed collection of spaces, and suppose K is a nilpotent space that can be built from spaces in 𝒜 by a succession of cofiber sequences. We show that, under mild conditions on the collection 𝒜, it is possible to construct K from spaces in 𝒜 using, instead, homotopy (inverse) limits and extensions by fibrations. One consequence is that if K is a nilpotent finite complex, then ΩK can be built from finite wedges of spheres using homotopy limits and extensions by fibrations. This is applied...

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