Higher Homotopy Groups Research Articles

Exploring Homotopy and Fundamental Groups: Definitions, Examples, and Key Propositions

This article delves into key concepts in algebraic topology, specifically focusing on homotopy, contractible spaces, fundamental groups, and simply connected spaces. Definitions, examples, and key propositions of these concepts are explored, providing insights into their mathematical foundations and applications. Homotopy, describing the continuous deformation between two objects, plays a crucial role in defining homotopy groups, which serve as important invariants in algebraic topology. The fundamental group, as the first homotopy group, is instrumental in analyzing the basic shape of a topological space, capturing information about its structure, such as the presence of holes. The paper also examines contractible spaces, which can be continuously reduced to a single point, and simply connected spaces, characterized by their trivial fundamental groups. Additionally, the article addresses advanced topics in homotopy theory, including fibrations, exact sequences, and higher homotopy groups, highlighting their importance in linking topological spaces and revealing complex homotopy relationships. The study emphasizes the relevance of homotopy theory in both mathematics and broader fields, such as physics, where it aids in visualizing the universe’s structure through analogies with fiber bundles.

Scalable spaces

Scalable spaces are simply connected compact manifolds or finite complexes whose real cohomology algebra embeds in their algebra of (flat) differential forms. This is a rational homotopy invariant property and all scalable spaces are formal; indeed, scalability can be thought of as a metric version of formality. They are also characterized by particularly nice behavior from the point of view of quantitative homotopy theory. Among other results, we show that spaces which are formal but not scalable provide counterexamples to Gromov’s long-standing conjecture on distortion in higher homotopy groups.

The infinitary n-cube shuffle

In this paper, we formalize the sense in which higher homotopy groups are "infinitely commutative." In particular, we both simplify and extend the highly technical procedure, due to Eda and Kawamura, for constructing homotopies that isotopically rearrange infinite configurations of disjoint $n$-cubes within the unit $n$-cube.

Derived ℓ-adic zeta functions

Abstract Let K 0 ( V k ) be the Grothendieck group of k-varieties. Campbell and Zakharevich have constructed a higher algebraic K-theory spectrum K ( V k ) such that π 0 K ( V k ) = K 0 ( V k ) . In this paper we construct non-trivial classes in the higher homotopy groups of K ( V k ) when k is finite or a subfield of C. To do this we give a recipe for lifting motivic measures K 0 ( V k ) → K 0 ( E ) to maps of spectra K ( V k ) → K ( E ) . We consider two special cases: the classical local zeta function, thought of as a homomorphism K 0 ( V F q ) → K 0 ( End ( Q l ) ) , and the compactly-supported Euler characteristic, thought of as a homomorphism K 0 ( V C ) → K 0 ( Q ) . We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map S → K ( V k ) is nontrivial in higher dimensions when k is finite or a subfield of C, and, moreover, that when k is finite this map is not surjective on higher homotopy groups.

Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groupspi _d(X) of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental grouppi _1(X) of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with pi _1(X) trivial), compute the higher homotopy group pi _d(X) for any given dge 2. However, these algorithms come with a caveat: They compute the isomorphism type of pi _d(X), dge 2 as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of pi _d(X). Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere S^d to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes pi _d(X) and represents its elements as simplicial maps from a suitable triangulation of the d-sphere S^d to X. For fixed d, the algorithm runs in time exponential in mathrm {size}(X), the number of simplices of X. Moreover, we prove that this is optimal: For every fixed dge 2, we construct a family of simply connected spaces X such that for any simplicial map representing a generator of pi _d(X), the size of the triangulation of S^d on which the map is defined, is exponential in mathrm {size}(X).

On the Principle of Least Action

Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle S=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path.

The annihilator of the Lefschetz motive

In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0 K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_0[\mathcal{V}_k]$ and show that classes in the kernel of multiplication by $[\mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are varieties such that $[X] \neq [Y]$, $X\times \mathbb{A}^1$ and $Y\times \mathbb{A}^1$ are not piecewise isomorphic, but $[X\times \mathbb{A}^1] =[Y\times \mathbb{A}^1]$ in $K_0[\mathcal{V}_k]$. Along the way we present new proofs of the result of Larsen--Lunts on the structure on $K_0[\mathcal{V}_k]/([\mathbb{A}^1])$.

Homological properties of determinantal arrangements

We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of 2 × n matrices satisfy nice combinatorial properties. We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of S 3 . Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincaré polynomials that factor nicely.

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.

ADJUNCTION OF mod P EQUIVALENCES IN HOMOTOPY THEORY

We formalize the adjunction of maps of localized nilpotent spaces of the homotopy type of CW-complexes. Using the semilocalization theory of M. Bendersky, in which only the higher homotopy groups are localized, we obtain an adjunction theorem with fewer nilpotency conditions on the spaces. The two main theorems are in the form of a theorem on maps of triads by J. P. May.

On the homotopy theory of complexes associated to metrical-hemisphere complexes

In the first section we introduce a certain class of sellular complexes, called metrical- hemisphere complexes (MH-complexes), which generalize oriented matroids, and study their main properties. In Section 2 another cellular complex is associated to each MH-complex; this construction generalizes that given in a preceding work. In Section 3 homotopy theory of the associated complexes is studied by introducing a map between a certain category of positive paths and the fundamental grupoid. If J is injective then the word problem for the fundamental group is solvable. In Sections 4, 5 higher homotopy groups are considered, giving sufficient criterions for the k (π, 1) property. Also, the results by Deligne concerning the simplicial arrangements is reproved in larger generality. In Section 6 the preceding results are exploited to produce new k (π, 1) spaces associated to arrangement of pseudo-hemispheres.

On the rationalization of the circle

We give an example showing that, for a nilpotent group G G and a set of primes P P , the P P -localization homomorphism l : G → G P l:G \to {G_P} need not induce an isomorphism in cohomology with arbitrary (twisted) Z P {{\mathbf {Z}}_P} -module coefficients. From this fact we infer that, in the pointed homotopy category of connected CW-complexes, the inclusion of the subcategory of spaces whose higher homotopy groups are Z P {{\mathbf {Z}}_P} -modules and whose fundamental group is uniquely P ′ {P’} -radicable does not admit a left adjoint.

The Behaviour of Homology in the Localization of Finite Groups

AbstractWe show that, for a finite group G and a prime p, the following facts are equivalent: (i) the p-localization homomorphism l: G —> Gp induces p-localization on integral homology; (ii) the higher homotopy groups of the Bousfield-Kan Zp-completion of a K(G, 1) vanish; (iii) the group G is p-nilpotent.

COMPUTING HOMOTOPY GROUPS OF A HOMOTOPY PULLBACK

Abstract Secondary structure of an exact sequence of Maya-Vietoris type associated to a simply-connected homotopy pullback is exploited to yield a technique for computing its higher homotopy groups. The information required consists of the homotopy groups of the original spaces, the homomorphisms induced by the given maps and certain matrix Toda brackets.

The Homotopy of Simplicial Algebras

In [6] Walter Taylor investigated the relationship between the algebraic structure of a topological algebra A and the group structure of its fundamental group π1(A) and of the higher homotopy groups πn(A),n > 1. The main result is that a variety satisfies a group law λ in homotopy (that is, π1) if and only if every group in the idempotent reduct of obeys λ. (The relevant definitions are in [6] and also § 2 of this paper.) A similar result is stated for the higher homotopy groups. As Taylor points out in the introduction, the hard part of the theorem is constructing a topological algebra in whose fundamental group may fail to obey λ; indeed, in [6] this is only done in detail for the commutative law, and the proof is rather computational.

Homology circles and knot complements

BY A lzonzology circle we mean a CW complex C (or a Kan complex) for which there is an isomorphism H,(C. 2) = H,(S’, 2) where S’ denotes the usual circle. Such spaces are always equipped with a map C + S’ which induces that isomorphism. The best known examples of homology circles are knot complements C = S”+‘N(K) where N(K) is an open tubular neighborhood of a submanifold K homeomorphic to the n-sphere S”. Here we are interested only in the homotop_v type of homology circles. Indeed it was proven by Browder, Lashof and Shaneson [9] that given the homotopy type of the pair (C, aC) it corresponds to, at most, two isotopy types of knots-if a,C = Z. From this point of view the homotopy type of (compact) homology circles is of major importance. Kervaire has done some fundamental work on the first non-trivial homotopy group ai(S”, C)[8] and we shall be interested in the higher homotopy groups. Kervaire also notes that “almost” arbitrary compact homology circles can be realized as knot complements, up to the middle dimension [see 5.12 below]. More generally, the present work is a continuation of the general work on homology isomorphism: How can one analyze and construct “all” homology circles? What are the possible homotopy groups and “k-invariants” of such spaces? What precise part of the (n + I)-type of C is determined by the n-type and what can be freely chosen? Such questions have relatively simple answers when one deals with e.g. acyclic spaces [3]. It is surprising to see that the case of homology circles turns out to be of the same general nature. But the details are more difficult since the fundamental group is not perfect. In the present paper we only deal with two types of homology circles: the (homotopy) finite case and the nilpotent case. We give a “classification” of these. It turns out that most of the important features of the general case arise already here and they assume here a concise and simple form. Roughly speaking the homotopy groups 7r” = TX appear naturally as extensions 0 + In, --, 7~, * nn IIT, + 0 where TV” is “a-perfect” and can be almost arbitrarily chosen and rr,,/I7r, depends on the (homologies of the) choices made in the lower dimension: In, for j <n. These considerations lead us to a general theorem which determines the structure of the second non-trivial homotopy group of a knot complement, in terms of the first non-trivial one (see 5.2). A word about the category in which we work: By “space” and “map” we mean objects and morphisms in the pointed category of CW complexes or Kan complexes. This small abuse of notation is harmless because we are interested in questions of homotopy type and the-geometric realizations provides an equivalence between the two categories [lo]. The paper is organized as follows. After formulating below the main results in general terms we recall some basic concepts and their properties in § I. $2 is devoted to the analysis of nilpotent homology circles. In 03 and 04 we explain and prove theorems relating to (homotopy) finite homology circles. Some applications and geometric remarks are given in $5.

Combinatorial Homotopy Theory and a New Proof that the Second Homotopy Group of the Circle is Trivial

I. We give here a new combinatorial definition for the second homotopy group of a simplicial complex. Though the definition generalizes easily to higher homotopy groups, it is convenient for the sake of notational simplicity to limit ourselves to the second homotopy group. The approach is less general and less sophisticated than that of D. M. Kan [1]. Using this definition we shall prove that the second homotopy group of the circle vanishes. We feel that the proof is interesting because of its extremely elementary nature using no topology or algebra! (The only place where topology would be required is in the proof of the equivalence of our definition of the homotopy groups and the usual one [2]. This is essentially a consequence of the simplicial approximation theorem.)

Homotopy-abelian Lie groups

where h* denotes the induced homomorphism. Note that #* is an isomorphism if A is a covering map and p, a ^ 2 . Hence if two topological groups have a common universal covering group then their higher homotopy groups are related by an isomorphism which is compatible with the Samelson product. Let <nrq(G), where g ^ l , denote the subset of w2q(G) consisting of elements ((3, fi), where fiCz7Tq(G). We assert the following

Some Homotopy Properties of Topological Groups and Homogeneous Spaces

Because of the continuous group operations, a topological group has some peculiar homotopy properties. It is well-known that the fundamental group of a topological group is commutative, [4, p. 54].2 The higher homotopy groups of a topological group have been investigated by W. Ifurewicz [11], B. Eckmann [5], and others. According to J. H. C. Whitehead [13], an arcwise connected topological group is n-simple for each n > 1 in the sense of S. Eilenberg [6]. According to R. H. Fox [81], the torus homotopy groups of a topological group G are all commutative; then it follows that the Whitehead products, [14], between the elements of the homotopy groups of G are always zero. The object of the present paper is to generalize these results to the class of special homogeneous spaces, defined in ?3, including the spheres Sn as particular cases. With an aim to do this, we shall give in ?2 some additional homotopy properties of a topological group. Our main results3 are embodied in the theorems of ?3. Throughout this paper, let G be a locally connected topological group. Since the components of G are homeomorphic to each other, we assume G to be arcwise connected. Let II be an arewise connected closed subgroup of G. A mapping is a continuous transformation, and a homotopy is a continuous family of mappings ft, (0 < t ? 1).

On the relation between the fundamental group on a space and the higher homotopy groups

On the relation between the fundamental group on a space and the higher homotopy groups