Search

In this work, we are concerned with the realization of spaces up to rational homotopy as classifying spaces. In this paper, we first show that a class of rank-two rational spaces cannot be realized up to rational homotopy as the classifying space of any n -connected and π -finite space for n ≥ 1. We also show that the Eilenberg-Mac Lane space K (Q r , n) (r ≥ 2 , n ≥ 2) can be realized up to rational homotopy as the classifying space of a simply connected and elliptic space X if and only if X has the rational homotopy type of ∏ r S n − 1 with n even. [ABSTRACT FROM AUTHOR]

In the paper we generalize the singular cubical homology theory of digraphs and introduce a collection of cell singular homologies that are parametrized by a natural parameter of the category of digraphs. The natural parameter corresponds to the "length" of a directed line digraph that corresponds to the unit segment in algebraic topology. The introduced earlier singular cubical homology is a particular case of this collection that corresponds to the line digraph with one arrow. We describe relations between these theories for different parameters given by a natural number and study their basic properties. In particular, we prove the functoriality and homotopy invariance of these theories and compare the theory with the path homology theory that was introduced in our previous papers. Then we describe the transfer of the cell singular homology theories to the categories of graphs, quivers, and multigraphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the exotic K (h) -local Picard groups κ h when 2 p − 1 = h 2 and the homological Chromatic Vanishing Conjecture when p − 1 does not divide h. The main idea is to use the Gross-Hopkins duality to relate both questions to certain Greek letter element computations in chromatic homotopy theory. Classical results of Miller-Ravenel-Wilson then imply that an exotic element at height 3 and prime 5 is not detected by the type-2 complex V (1). For the homological Vanishing Conjecture, we prove it holds modulo the invariant prime ideal I h − 1. We further show that this special case of the Vanishing Conjecture implies the exotic Picard group κ h is zero at height 3 and prime 5. Both results can be thought of as a first step towards proving the vanishing of κ 3 at prime 5. [ABSTRACT FROM AUTHOR]

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variation of the model has better properties: it is regular and sits inside the configuration space as a strong and equivariant deformation retract. Our variant for the configuration-space model comes from a comparison, in the equivariant setting, between Wiltshire's simplicial difference and a well known model for the complement of a full subcomplex on a simplicial complex. [ABSTRACT FROM AUTHOR]

We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc (K) = 4 , where K denotes Klein bottle. [ABSTRACT FROM AUTHOR]

Abstract We characterize the Hurewicz cofibrations between finite topological spaces, that is, the continuous functions between finite topological spaces that have the homotopy extension property with respect to all topological spaces. In particular, we show that cofibrations between connected non-empty finite topological spaces are homotopy equivalences. As a consequence of our characterization, we obtain a simple algorithm capable of determining whether a given continuous function between finite topological spaces is a cofibration. [ABSTRACT FROM AUTHOR]

Abstract Let M l , m be the total space of the S 3 -bundle over S 4 classified by the element l σ + m ρ ∈ π 4 (S O (4)) , l , m ∈ Z. In this paper we study the homotopy theory of gauge groups of principal G -bundles over manifolds M l , m when G is a simply connected simple compact Lie group such that π 6 (G) = 0. That is, G is one of the following groups: S U (n) (n ≥ 4) , S p (n) (n ≥ 2) , S p i n (n) (n ≥ 5) , F 4 , E 6 , E 7 , E 8. If the integral homology of M l , m is torsion-free, we describe the homotopy type of the gauge groups over M l , m as products of recognisable spaces. For any manifold M l , m with non-torsion-free homology, we give a p -local homotopy decomposition, for a prime p ≥ 5 , of the loop space of the gauge groups. [ABSTRACT FROM AUTHOR]

Abstract We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data. [ABSTRACT FROM AUTHOR]

Abstract We study some Borsuk–Ulam type results for the loop space of an euclidean sphere without loops equal to their inverses. [ABSTRACT FROM AUTHOR]

Abstract In 1995 Heath, Keppelmann and Wong studied a Nielsen type number N F (f , p) for a fibre preserving map f of fibration p. This number is a lower bound for the least number of fixed points within the fibre homotopy class of f. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers N P n F (f , p) and N Φ n F (f , p) for periodic points of a fibre preserving map. These numbers, which can be thought of as the dual of periodic Nielsen type numbers for a relative map due to Heath, Schirmer and You, are bigger than the ordinary Nielsen type numbers. The definition of N P n F (f , p) and N Φ n F (f , p) is reminiscent of the naive type addition formulae for periodic points of fibre preserving maps due to Heath and Keppelmann. Some calculations and examples are given. [ABSTRACT FROM AUTHOR]

Abstract Let M be a smooth manifold, and let O (M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F : O (M) ⟶ Spaces from O (M) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤ k are determined by their values on O k (M) , where O k (M) is the full subposet of O (M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace O k (M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O (M) to any simplicial model category M. If F k (M) stands for the unordered configuration space of k points in M , we also show that the category of homogeneous cofunctors O (M) ⟶ M of degree k is weakly equivalent to the category of linear cofunctors O (F k (M)) ⟶ M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F : O k (M) ⟶ M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion O k (M) ↪ O (M) is also an isotopy cofunctor. [ABSTRACT FROM AUTHOR]

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of wedge sum of finitely many spheres of various dimensions. [ABSTRACT FROM AUTHOR]

We determine the group structure of the 23-rd homotopy group π 23 ( G 2 ) , where G 2 is the exceptional Lie group of rank 2, which hasn't been determined for 50 years. [ABSTRACT FROM AUTHOR]

In this paper every polyhedron is finite and every ANR is compact. Let P ≥ X 1 ≥ X 2 ≥ … , be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer l P (independent of the sequence) that each sequence contains only ≤ l P homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra { P i } , where P i homotopy dominates P i + 1 but P i and P i + 1 have different homotopy types, for every i ∈ N ? (M. Moron) [30, Problem 1436] , and the famous problem of K. Borsuk (1967): Is it true that two A N R ′ s homotopy dominating each other have the same homotopy type? Here we prove that, if dim ⁡ P = 2 , then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim ⁡ P ≤ 2 and elementary amenable fundamental group G with cd G < ∞ , there is a bound l P on the lengths of all descending sequences P ≥ X 1 ≥ X 2 … of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth. [ABSTRACT FROM AUTHOR]

It is well-known that a paracompact space X is of covering dimension at most n if and only if any map f : X → K from X to a simplicial complex K can be pushed into its n -skeleton K ( n ) . We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map f is replaced by variation of f on elements of a uniformly bounded cover. In the same way, one can generalize Property A of G. Yu to arbitrary coarse spaces. [ABSTRACT FROM AUTHOR]

Let G k denote the gauge group of the principal S p ( 3 ) -bundle over S 4 with first symplectic Pontryagin class k ∈ H 4 S 4 = Z . We show that when localised at an odd prime there is a homotopy equivalence G k ≃ G l if and only if ( 21 , k ) = ( 21 , l ) . We also give bounds on the number of 2-local homotopy types amongst the G k . Our results follow from a thorough examination of the commutator map c : S p ( 1 ) ∧ S p ( 3 ) → S p ( 3 ) and we determine its order to be either 4 ⋅ 3 ⋅ 7 = 84 , 8 ⋅ 3 ⋅ 7 = 168 or 16 ⋅ 3 ⋅ 7 = 336 . In particular its order at odd primes is completely determined. [ABSTRACT FROM AUTHOR]

The notion of a derived A -infinity algebra, considered by Sagave, is a generalization of the classical notion of A -infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r -homotopy, for non-negative integers r , in such a way that r -homotopy equivalences underlie E r -quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A -infinity algebras as A -infinity algebras in twisted complexes and as A -infinity algebras in split filtered cochain complexes. [ABSTRACT FROM AUTHOR]

By using the notion of polyhedral products ( X _ , A _ ) K , we recognise the Bestvina–Brady construction [4] as the fundamental group of the homotopy fibre of ( S 1 , ⁎ ) L → S 1 , where L is a flag complex. We generalise their construction by studying the homotopy fibre F of ( S 1 , ⁎ ) L → ( S 1 , ⁎ ) K for an arbitrary simplicial complex L and K an ( m − 1 ) -dimensional simplex. For a particular class of simplicial complexes L , we describe the homology of F , its fixed points, and maximal invariant quotients for coordinate subgroups of Z m . This generalises the work of Leary and Saadetoğlu [13] who studied the case when m = 1 . [ABSTRACT FROM AUTHOR]

Let G be a compact Lie group. (Compact) topological G -manifolds have the G -homotopy type of (finite-dimensional) countable G -CW complexes (2.5) . This partly generalizes Elfving's theorem for locally linear G -manifolds [14] , wherein the Lie group G is linear (such as compact). [ABSTRACT FROM AUTHOR]

We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization [2] . For an operad O and a sub-monoid of one-ary operations W we associate an operad L O and a canonical map O → L O which takes elements in W to homotopy invertible operations. Furthermore, we give a functor from the category of O -algebras to the category of L O -algebras satisfying an appropriate universal property. [ABSTRACT FROM AUTHOR]

We establish a formal framework for Rognes's homotopical Galois theory and adapt it to the context of motivic spaces and spectra. We discuss examples of Galois extensions between Eilenberg–MacLane motivic spectra and between the Hermitian and algebraic K -theory spectra. [ABSTRACT FROM AUTHOR]

For integers m , n , d ≥ 1 with ( m , n ) ≠ ( 1 , 1 ) and a field F with algebraic closure F ‾ , let Poly d , m n ( F ) denote the space of all m -tuples ( f 1 ( z ) , … , f m ( z ) ) ∈ F [ z ] m of monic polynomials of the same degree d such that polynomials f 1 ( z ) , … , f m ( z ) have no common root in F ‾ of multiplicity ≥ n . These spaces were defined by Farb and Wolfson [12] as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. They obtained algebraic geometrical and arithmetic results about them. In this paper we investigate the homotopy type of these spaces for the case F = C . Our results generalize those of [12] for F = C and also results of G. Segal [28] , V. Vassiliev [30] and F. Cohen–R. Cohen–B. Mann–R. Milgram [5] for m ≥ 2 and n ≥ 2 . [ABSTRACT FROM AUTHOR]

We develop a new method in the computation of equivariant homotopy, which is based on the splitting of cofiber sequences associated to universal spaces in the category of equivariant spectra. In particular, we use this method to compute the homotopy of H Z _ for G = D 2 p and A 5. • A new computational method in equivariant stable homotopy theory: The core is stated in Theorem 4.6. • Discussion on the computability of H Z _ and M U G (sections 4.3 and 4.4). • Explicit computation on π ★ G (H Z _) for G = D 2 p and A 5 (sections 5 and 6). [ABSTRACT FROM AUTHOR]

We generalize Brazas' topology on the fundamental group to the whole universal path space X ˜ , i.e., to the set of homotopy classes of all based paths. We develop basic properties of the new notion and provide a complete comparison of the obtained topology with the established topologies, in particular with the Lasso topology and the CO topology, i.e., the topology that is induced by the compact-open topology. It turns out that the new topology is the finest topology contained in the CO topology, for which the action of the fundamental group on the universal path space is a continuous group action. [ABSTRACT FROM AUTHOR]

Let ( V , W ; F ) be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold M . Then M o d ( M , F ) naturally acts on the disk complex D ( F ) as a group action. In this article, we prove if F is topologically minimal and its topological index is two, then the orbit of any element of D ( F ) for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of D ( F ) whose orbits are finite if the genus of F is three. [ABSTRACT FROM AUTHOR]

Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim ⁡ X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G ⁡ X ≤ n . [ABSTRACT FROM AUTHOR]

The equivalence class of a principal G 2 -bundle over S 4 is classified by the value k ∈ Z of the second Chern class. In this paper we consider the homotopy types of the corresponding gauge groups G k , and determine the number of homotopy types up to one factor of 2. [ABSTRACT FROM AUTHOR]

Some relations between self-homotopy equivalences of a space and those of the induced cofiber are obtained. The results are applied to prove relations between self-closeness numbers of spaces and their mapping cones. Furthermore, various results are obtained for some types of rational spaces making use of models of Sullivan and Quillen. [ABSTRACT FROM AUTHOR]

We consider 32 homotopy classifications of knot projections (images of generic immersions from a circle into a 2-sphere). These 32 equivalence relations are obtained based on which moves are forbidden among the five types of Reidemeister moves. We show that 32 cases contain 20 non-trivial cases that are mutually different. To complete the proof, we obtain new tools, i.e., new invariants. [ABSTRACT FROM AUTHOR]

We completely determine spherical classes in single, double and triple loop spaces of spheres. We also show that for l ⩾ 4 and n > 1 , there exists no spherical class in H ⁎ Ω l S l + n in dimensions more than 2 l n + 2 l − 1 ( l − 2 ) + 2 . This bound may not be optimal, but reduces the computation of spherical classes in H ⁎ Ω l S l + n to verifying finite number of cases. We also provide a table for possible spherical classes in H ⁎ Ω l S l + n with 3 < l < 9 . Our results, provide positive evidence for the truth of Eccles conjecture. [ABSTRACT FROM AUTHOR]

In 1958, Smale proved that any immersions of S 2 to R 3 are regularly homotopic. This means that we can turn an embedded sphere in R 3 inside out by a regular homotopy. After Smale showed his result without visualization, many people visualized sphere eversions, in various ways. In this paper, we construct a sphere eversion by lifting a “simple” generic homotopy of S 2 to R 2 to a generic regular homotopy of S 2 to R 3 . By doing so, our eversion is simple in terms of deformation of the contour generators of immersed spheres. We also visualize the 3-dimensional interlinking of the contour generators and the self-intersections of each stage of immersed spheres. [ABSTRACT FROM AUTHOR]

Viewing Kan complexes as ∞-groupoids implies that pointed and connected Kan complexes are to be viewed as ∞-groups. A fundamental question is then: to what extent can one “do group theory” with these objects? In this paper we develop a notion of a finite ∞ -group : an ∞-group whose homotopy groups are all finite. We prove a homotopical analogue of Sylow theorems for finite ∞-groups. This theorem has two corollaries: the first is a homotopical analogue of Burnside's fixed point lemma for p -groups and the second is a “group-theoretic” characterisation of finite nilpotent spaces. [ABSTRACT FROM AUTHOR]

We investigate the classical Alexandroff–Borsuk problem in the category of non-triangulable manifolds: Given an n -dimensional compact non-triangulable manifold M n and ε > 0 , does there exist an ε -map of M n onto an n -dimensional finite polyhedron which induces a homotopy equivalence? [ABSTRACT FROM AUTHOR]

Suppose X and Y are compact connected topological 4-manifolds with fundamental group π . For any r ⩾ 0 , X is r -stably homeomorphic to Y if X # r ( S 2 × S 2 ) is homeomorphic to Y # r ( S 2 × S 2 ) . How close is stable homeomorphism to homeomorphism? When the common fundamental group π is virtually abelian, we show that large r can be diminished to n + 2 , where π has a finite-index subgroup that is free-abelian of rank n . In particular, if π is finite then n = 0 , hence X and Y are 2-stably homeomorphic, which is one S 2 × S 2 summand in excess of the cancellation theorem of Hambleton–Kreck [12] . The last section is a case study of the homeomorphism classification of closed manifolds in the tangential homotopy type of X = X − # X + , where X ± are closed nonorientable topological 4-manifolds with order-two fundamental groups [13] . [ABSTRACT FROM AUTHOR]

We introduce a new lower-Vietoris-type hypertopology in a way similar to that with which a new upper-Vietoris-type hypertopology was introduced in [4] (it was called there Tychonoff-type hypertopology ). We study this new hypertopology and, in particular, we generalize many results from [3] . As a corollary, we get that for every continuous map f : X ⟶ X , where X is a continuum, there exist a subcontinuum K of X such that f ( K ) = K . [ABSTRACT FROM AUTHOR]

The purpose of this article is to compute the cohomology of B Diff ( K ) , the classifying space of the diffeomorphisms of the Klein bottle. We analyze the Serre spectral sequence of the fibration B Diff 0 ( K ) → B Diff ( K ) → B ( Z 2 × Z 2 ) , with non-trivial coefficients, and show it collapses. Moreover, we show that B Diff ( K ) has the homotopy type of B Z 2 × B O ( 2 ) . Finally we give direct and concrete constructions for the Eilenberg–MacLane spaces K ( Γ q ( K ) , 1 ) , where Γ q ( K ) denotes the mapping class group of K with marked points π 0 Diff ( K ; q ) . [ABSTRACT FROM AUTHOR]

We compute the Bousfield localizations and Bousfield colocalizations of discrete model categories, including the homotopy categories and the algebraic K -groups of these localizations and colocalizations. We prove necessary and sufficient conditions for a subcategory of a category to appear as the subcategory of fibrant objects for some such model structure. We also prove necessary and sufficient conditions for a monad to be the fibrant replacement monad of some such model structure. [ABSTRACT FROM AUTHOR]

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most d , for d ≥ 2 . We denote these spaces by O d , P d and Q d . For d ≥ 3 , we show that the spaces O d and P d are path connected and the space O d has the same homotopy type as S 2 . Considering the space P = ⋃ d ≥ 2 O d of all polynomial knots with the inductive limit topology, we prove that it too has the same homotopy type as S 2 . We also show that if two polynomial knots are path equivalent in Q d , then they are topologically equivalent. Furthermore, the number of path components in Q d are in multiples of eight. [ABSTRACT FROM AUTHOR]

Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration that mimics the smooth setting. [ABSTRACT FROM AUTHOR]

We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation. [ABSTRACT FROM AUTHOR]

We determine an upper bound for the number of homotopy associative multiplications on certain H -spaces. This is applied to S U ( 3 ) and S p ( 2 ) at odd primes, and to give examples of p -local H -spaces with more than one multiplication but a unique homotopy associative multiplication. [ABSTRACT FROM AUTHOR]

The Lipshitz-Sarkar Steenrod operations on (even) Khovanov homology are incompatible with Szabó's geometric spectral sequence. Obstructions to integral lifting and spectrification are observed. [ABSTRACT FROM AUTHOR]

In the article we have introduced a notion of relative homotopy. Thus we have recalled a notion of relative retract and relative extension of continuous maps. We have studied some properties of a relative homotopy and we have proven that it is an equivalence relation. We have defined the notion of relative contractibility and we have proven the theorem on the extension of a relative homotopy. We have shown that a relative homotopy of continuous maps has similar properties to a homotopy of continuous maps. We have given a few applications of a relative homotopy that extend our knowledge of the fixed point theory, the theory of coincidence and the topological properties of acyclic sets. [ABSTRACT FROM AUTHOR]

In 2001, Östlund formulated the question: are Reidemeister moves of types 1 and 3 sufficient to describe a homotopy from any generic immersion of a circle in a two-dimensional plane to an embedding of the circle? The positive answer to this question was treated as a conjecture (Östlund conjecture). In 2014, Hagge and Yazinski disproved the conjecture by showing the first counterexample with a minimal crossing number of 16. This example is naturally extended to counterexamples with given even minimal crossing numbers more than 14. This paper obtains the first counterexample with a minimal crossing number of 15. This example is naturally extended to counterexamples with given odd minimal crossing numbers more than 13. [ABSTRACT FROM AUTHOR]

A class of pointed spaces is called a resolving class if it is closed under weak equivalences and pointed homotopy limits. Let R ( A ) denote the smallest resolving class containing a space A . We say X is A -resolvable if X is in R ( A ) , which induces a partial order on the category pointed spaces. We develop an algebraic criterion for determining if X is A -resolvable when X and A are rational spaces. The goal of this work is to develop some understanding of the structure of this resolvability relation, and in particular to use our algebraic criterion to help us better understand the rational case. [ABSTRACT FROM AUTHOR]

2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. We develop an explicit efficient algorithm to decide when a 2-stratifold is as in the title. [ABSTRACT FROM AUTHOR]

The configuration space F 2 ( M ) of ordered pairs of distinct points in a manifold M , also known as the deleted square of M , is not a homotopy invariant of M : Longoni and Salvatore produced examples of homotopy equivalent lens spaces M and N of dimension three for which F 2 ( M ) and F 2 ( N ) are not homotopy equivalent. They also asked whether two arbitrary 3-dimensional lens spaces M and N must be homeomorphic in order for F 2 ( M ) and F 2 ( N ) to be homotopy equivalent. We give a partial answer to this question using a novel approach with the Cheeger–Simons differential characters. [ABSTRACT FROM AUTHOR]

We introduce the right homotopy shift of paths and loops in the inverse limit of a single upper semi-continuous multivalued function on the unit interval. Consequently we obtain a shift in the fundamental group, which turns out to be an injective map under conditions usually assumed for such one-dimensional limits. As a result we obtain strong restriction on the fundamental groups of such inverse limits: they are not finitely generated and are often trivial or uncountable. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the topology of certain toric varieties X I , arising as quotients of the action of C ⁎ on complements of arrangements of coordinate subspaces in C n , and to improve the homotopy stability dimension for the inclusion map i d : Hol d ⁎ ( S 2 , X I ) → Map d ⁎ ( S 2 , X I ) given in [11] by making use of the Vassiliev spectral sequence. We also improve the homotopy stability dimension of this inclusion given by G. Segal [18] for X I = C P n − 1 and n ≥ 3 . [ABSTRACT FROM AUTHOR]

We show that relative loop spaces are recognized by A ∞ -actions. A certain version of the 2-sided bar construction is used to prove such recognition theorem. The operad Act ∞ of A ∞ -actions is presented in terms of the Boardman–Vogt resolution of the operad Act . We exhibit an operad homotopy equivalence between such resolution and the 1-dimensional Swiss-cheese operad SC 1 . [ABSTRACT FROM AUTHOR]