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We construct algebraic unoriented and oriented cobordism, named MGLO and MSLO , respectively. MGLO is defined and its homotopy groups are explicitly computed, giving an answer to a question of Jack Morava. MSLO is also defined and its coefficients are explicitly computed after completing at a prime p . Similarly to MSO , the homotopy type of MSLO depends on whether the prime p is even or odd. Finally, a computation of a localization of the homotopy groups of MGLR is given.

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner, the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterized in terms of congruences. This is first shown to be a stable problem and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe., Comment: 20 pages

A so-called special generic map is by definition a map of smooth manifolds all of whose singularities are definite fold points. It is in general an open problem posed by Saeki in 1993 to determine the set of integers $p$ for which a given homotopy sphere admits a special generic map into $\mathbb{R}^{p}$. By means of the technique of Stein factorization we introduce and study certain special generic maps of homotopy spheres into Euclidean spaces called standard. Modifying a construction due to Weiss, we show that standard special generic maps give naturally rise to a filtration of the group of homotopy spheres by subgroups that is strongly related to the Gromoll filtration. Finally, we apply our result to concrete homotopy spheres, which particularly answers Saeki's problem for the Milnor $7$-sphere., 11 pages; essential application to Milnor 7-sphere added, some minor typos eliminated

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.

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.

In this paper we study notions of homotopy in the Newtonian space N1,p(X;Y) of Sobolev type maps between metric spaces. After studying the properties and relations of two different notions we prove a compactness result for sequences in homotopy classes with controlled homotopies.

We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a “thickening” of the usual category of supermanifolds; it is close to the category of symplectic micromanifolds and their micromorphisms considered recently by A. Weinstein and A. Cattaneo–B. Dherin–A. Weinstein.) There are two parallel settings, for even and odd functions. As an application, we show how such nonlinear pullbacks give L ∞ -morphisms for algebras of functions on homotopy Schouten or homotopy Poisson manifolds.

The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory (HoTT) research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising.

Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected., Comment: 9 pages

In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper, we prove that the same problem for $Y$ an odd-dimensional sphere is decidable. More generally, the same holds for any $d$-connected target space $Y$ whose homotopy groups $\pi_k Y$ are finite for $k>2d$., Comment: 6 pages

We determine the 2-primary components of the 32-stem homotopy groups of spheres. The method is based on the classical one including the Toda’s composition methods.

Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend lower categorical analogues that have been classically used in algebraic topology and algebraic K-theory, such as the homotopy invariance theorem (by Bousfield and Kan), the homotopy colimit theorem (Thomason), Theorems A and B (Quillen), or the homotopy cofinality theorem (Hirschhorn).

In this paper we generalize the retracting property in homotopy theory for topological semigroups by introducing the notions of deformation S-retraction with its weaker forms and ES-homotopy extension property. Furthermore, the covering homotopy theorems for S-maps into Sχ-fibrations and Sχ-cofibrations are introduced and pullbacks for Sχ-fibrations behave properly.

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.

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.

We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

We study invariants of regular immersions of graphs into surfaces up to regular homotopy. The concept of the winding number is used to introduce a new simple combinatorial invariant of regular homotopy. Bibliography: 20 titles.

We study the stable motivic homotopy groups $\pi_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an $\eta$-local region that is entirely known, a $\tau$-local region that is identical to classical stable homotopy groups, and a region that is not well-understood., Comment: 8 pages, 1 figure

We prove that homotopy invariants of finite degree distinguish homotopy classes of maps of a connected compact CW-complex to a nilpotent connected CW-complex with finitely generated homotopy groups. Bibliography: 12 titles.

A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engstr\"om recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number of faces in the Engstr\"om representation corresponding to a codimension one homotopy sphere arrangement is bounded above by a polynomial in the number of elements of the matroid with degree one less than its rank., Comment: 9 pages, 4 figures

The study of n-dimensional manifolds has seen great advances in the last half century. In dimensions greater than four, surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial, where serious algebraic issues remain. In dimension 3, the proof by Perelman of Thurston’s Geometrization Conjecture (1) allows an algorithmic classification of 3-manifolds. The work of Freedman (2) classifies topological 4-manifolds if the fundamental group is not too large. Also, gauge theory in the hands of Donaldson (3) has provided invariants leading to proofs that some topological 4-manifolds have no smooth structure, that many compact 4-manifolds have countably many smooth structures, and that many noncompact 4-manifolds, in particular 4D Euclidean space R 4 , have uncountably many. However, the gauge theory invariants run into trouble with small 4-manifolds, such as those with the same homology groups as the 4D sphere, S 4 . In particular, the smooth 4D Poincare Conjecture, the last remaining case of that hallowed conjecture, is still open. (In higher dimensions, the smooth Poincare Conjecture is sometimes true in the following sense. In dimensions 3, 5, 6, 12, and 61, a homotopy sphere is diffeomorphic to the standard one, and in all other known cases, there are increasingly many exotic smooth structures on the topological sphere; however, it is possible that there may be more high-dimensional cases with no exotic spheres.) The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic. What is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions. The s-cobordism theorem states that, if M 0 m and M 1 m are the two boundary components of an m + 1 -dimensional manifold W and if … [↵][1]1Email: kirby{at}math.berkeley.edu. [1]: #xref-corresp-1-1

A link tower is a sequence of links with the structure given by removing the last components. Given a link tower, we prove that there is a chain complex consisting of (non-abelian) groups given by the symmetric commutator subgroup of the normal closures in the link group of the meridians excluding the meridian of the last component with the differential induced by removing the last component. Moreover, the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis. These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jollenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z_K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop space of Z_K is homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime not equal to 2.

In this paper, we construct a framework which is called the digital homotopy fixed point theory. We get new results associating digital homotopy and fixed point theory. We also give an application on this theory.

Let S(V) be a complex linear sphere of a finite group G. Let S(V)*n denote the n-fold join of S(V) with itself and let aut G(S(V)*) denote the space of G-equivariant self-homotopy equivalences of S(V)*n. We show that for any k ≥ 1 there exists M > 0 that depends only on V such that |πk autG(S(V)*n)|≤ M for all n ≫0.

This paper is devoted to prove several results concerning the homotopy groups of separable metric spaces that generalize some of the main results of [4] and [5] to homotopy groups. In particular, we focus on subspaces of Euclidean spaces. Among the results, we proposed a partial generalization of Shelah's Theorem to higher homotopy groups for noncompact spaces. Also, we discuss n -homotopically Hausdorff property, a separation axiom for n -loops introduced in [12] , and conclude that each subset of R n + 1 is n -homotopically Hausdorff. Moreover, the concept of a Hawaiian n -wild point will be introduced that illustrates the complexity of homotopy group at that point. We show that any ( n − 1 ) -connected locally ( n − 1 ) -connected subspaces of R n + 1 with uncountable n th homotopy group admit a Hawaiian n -wild point. Finally, we prove that n th homotopy group of any ( n − 1 ) -connected locally ( n − 1 ) -connected subspace of R n + 1 is free provided that it is n -semilocally simply connected, and then we study the free Abelian factor groups of the homotopy groups of these spaces.

Our goal in this paper is to give a full classification of the rational homotopy type of any elliptic and simply connected space when the sum of its Betti numbers is less or equal than 8.

We generalize the homotopy exact sequences of étale fundamental groups for proper separable fibrations to the case where fibrations are not necessarily proper and separable. To treat the case where fibrations admit nonreduced geometric fibers, we introduce orbifolds within the framework of schemes and study their fundamental groups. As an application, we give a criterion for simple-connectedness of elliptic surfaces over an algebraically closed field by classifying simply connected orbifold curves.

This is a survey of central results in nonabelian algebraic topology. We present how the homotopy category of homotopy $$n$$ -types and a certain localization of the category of crossed $$n$$ -cubes of groups are equivalent. The functor inducing this equivalence satisfy a generalized Seifert-van Kampen theorem, in that it preserves connectivity and colimits of certain diagrams of generalized fibrations. We show descriptions of certain colimits of crossed $$n$$ -cubes of groups and show how they have been used to generalize the Blakers-Massey theorem, the Hurewicz theorem and Hopf’s formula for the homology of groups, as well as a combinatorial formula for the homotopy groups of the sphere $$\mathbb {S}^2$$ . We also study the wedge sum of Eilenberg-MacLane spaces.

Algebraic objects arising from the study of curves and graphs on 2-surfaces are studied. Their homotopy invariance is verified.

In this paper we study the homotopy rigidity property of the functors ΣΩ and Ω. Our main result is that both functors are homotopy rigid on simply-connected p-local finite co-H-spaces. The result is obtain by a subtle interplay of homotopy decomposition techniques, modular representation theory and the counting principle.