On homotopy varieties

Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties.     

Fundamental group of homotopy colimits

We give a general version of theorems due to Seifert-van Kampen and Brown about the fundamental group of topological spaces. We consider here the fundamental group of a general homotopy colimit of spaces. This includes unions, direct limits and quotient spaces as special cases. The fundamental group of the homotopy colimit is determined by the induced diagram of fundamental groupoids via a simple commutation formula. We use this framework to discuss homotopy (co-)limits of groups and groupoids as well as the useful Classification Lemma 6.4. Immediate consequences include the fundamental group of a quotient spaces by a group action and of more general colimits. The Bass-Serre and Haefliger's decompositions of groups acting on simplicial complexes is shown to follow effortlessly. An algebraic notion of the homotopy colimit of a diagram of groups is treated in some detail.     

Pairings of function spaces

Pairings and copairings of topological spaces induce pairings of function spaces. These induced pairings of function spaces are studied. For this purpose, the C-open topology of function spaces is studied for subcategories C of Top. It is shown that the C-open topology enjoys good properties for homotopy theory. Making use of the C-open topology, theory of induced pairings is established and fundamental results on pairings which deduce various commutativity properties of elements in homotopy set are extended to function spaces.     

关于同伦正则态射

该文在点标拓扑空间的范畴中引进了同伦正则态射的概念,研究了它存在的条件,性质以及它与同伦单（满）态,同伦正则单（满）态和同伦等价之间的密切关系。     

Parametrized spaces model locally constant homotopy sheaves

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π₁-sets. We then use these two equivalences to study base change functors for parametrized spaces.     

Compact open topology and CW homotopy type

The class of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space and a compact Hausdorff space X, the space Y^X of continuous functions X→Y, endowed with the compact open topology, belongs to . P.J. Kahn extended this in 1982, showing that if X has finite n-skeleton and π_k(Y)=0, k>n.

Using a different approach, we obtain a further generalization and give interesting examples of function spaces where is not homotopy equivalent to a finite complex, and has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type.

As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.     

Fixed points and graph images for minimal maps on the once punctured torus

A fixed point detection theorem for a family of maps defined on the once punctured torus is proved. As a consequence, we produce an example of a homotopy class [f] of self-maps on the once punctured torus that illustrates the following: (i) there is a map in the homotopy class that has no fixed points, and (ii) if the image of f lies in a 1-complex that embeds as a homotopy equivalence, then f must have a fixed point.     

Rational Homotopy Theory and Nonnegative Curvature

in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit （complete） metrics with non-negative curvature.     

The coherent homotopy category over a fixed space is a category of fractions

In this note we prove that the coherent homotopy category over a fixed space B with morphisms represented by certain homotopy commutative squares (see [8]) is isomorphic to the category obtained by formally inverting those maps in the category Top_B of topological spaces over B which are ordinary homotopy equivalences.     

Strong homotopy induced by adjacency structure

This paper generalizes the concept of SA-homotopy in finite topological adjacency category, which is introduced in our previous work, to graph category and discusses its properties. We prove that every SA-strong deformation retract of a simple graph G could be obtained by removing trivial vertices one by one, which makes it possible to allow an iterative algorithm of finding all SA-strong deformation retracts of G. We also obtain that two simple graphs are SA-homotopy equivalent if and only if they have graph isomorphic cores. Compared with the graph homotopy transformation defined by S.T. Yau et al. and the s-homotopy transformation defined by R. Boulet et al., the main advantage of SA-homotopy transformation is that it could reflect correspondences between vertices, and hence it more accurately describe the transformation process than the graph homotopy transformation and s-homotopy transformation. As an application of SA-homotopy on graphs, we introduce the mapping class group of a graph, which also shows its advantage over the graph homotopy transformation and the s-homotopy transformation.     

Commutativity in special unitary groups at odd primes

It is a classical result by Bott that SU(s) and SU(t) homotopy commute in SU(n) if and only if s+t?n. We consider the p-localization analog of this problem and give an answer at odd primes.     

Regenerative cascade homotopies for solving polynomial systems

A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.     

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

Modelling and computing homotopy types: I

The aim of this article is to explain a philosophy for applying the 1-dimensional and higher dimensional Seifert–van Kampen Theorems, and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at the border between homology and homotopy. We explain some applications to filtered spaces, and special cases of them, while a sequel will show the relevance to $n$-cubes of pointed spaces.     

Absolute extensors and absolute neighborhood extensors in asymptotic categories

This paper deals with extending maps in asymptotic categories, i.e., in categories consisting of metric spaces and asymptotically Lipschitz coarsely proper maps. We demonstrate certain examples of absolute extensors and absolute neighborhood extensors. We give some conditions under which a version of Borsuk's homotopy extension theorem holds in these categories, and in answer to a problem posed by Dranishnikov in [Russian Math. Surveys 55 (2000) 1085] we show the failure of a general homotopy extension theorem. Finally, we show that a pair of an Hadamard space and its convex subspace has the homotopy extension property.     

Rational homotopy theory for non-simply connected spaces

We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.

     

Calculus II: Analytic functors

Homotopy functors (for example, from spaces to spaces) are called analytic if, when evaluated on certain n-cubical diagrams, they satisfy certain connectivity estimates. Tools for verifying these estimates include certain generalizations of the triad connectivity theorem. Waldhausen's functor A is analytic. Analyticity has strong consequences, when combined with the concept derivative of a homotopy functor that was introduced in the previous article in this series. In particular, any analytic functor with derivative zero is, in a sense, locally constant.Research partially supported by NSF grant DMS-8806444 and a Sloan Fellowship.