We provide a general definition of higher homotopy operations, encompassing most known cases, including higher Massey and Whitehead products, and long Toda brackets. These operations are defined in terms of the W-construction of Boardman and Vogt, applied to the appropriate diagram category; we also show how some classical families of polyhedra (including simplices, cubes, associahedra, and permutahedra) arise in this way.
In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.
The first aim of this paper is to study the $p$-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the $p$-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the $5$-local higher homotopy non-commutativity problem of the exceptional Lie group $\mathrm{G}_2$, which has been open for a long time.
We describe the first stem of the stable homotopy groups of spheres by using some Puppe sequences, Thom complexes, K-Theory and Adams operations following the ideas of J. Frank Adams. We also touch upon the second and the third stems in this perspective. Neither spectral squences nor Steenrod operations are used.
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $X$ up to weak equivalence. It can also be used to distinguish between different maps $f$ from $X$ to $Y$ which induce the same morphism on homotopy groups $f_*$ from $π_* X$ to $π_* Y$.
Let $(G,H)=(SU(2n+1),SO(2n+1)),\,(SU(2n),Sp(n)),\,(SO(2n),SO(2n-1)),\,(E_6,F_4),\,(Spin(8),G_2)$, and let $p$ be any prime $\ge 5$ for $(G,H)=(E_6,F_4)$, any prime $p\ne 3$ for $(G,H)=(Spin(8),G_2)$, and any odd prime otherwise. The classical result of Harris on the relation between the homotopy groups of $G$ and $H$ is reinterpreted as a $p$-local homotopy equivalence $G\simeq_{(p)}H\times G/H$, which yields a projection $G_{(p)}\to H_{(p)}$. We show how much this projection preserves the higher homotopy associativity.
The purpose of this paper is to introduce $N_k(\ell)$-maps ($1\le k,\ell\le\infty$), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_k(\ell)$-map is defined with higher homotopical conditions. It is shown that a homomorphism is an $N_k(\ell)$-map if and only if there exists fiberwise maps between fiberwise projective spaces with some properties. Also, the homotopy quotient of an $N_k(k)$-map is shown to be an $H$-space if its LS category is not greater than $k$. As an application, we investigate when the inclusions $\operatorname{SU}(m)\to\op…
In this note we show that the second homotopy group of $B(2,G)$, the classifying space for commutativity for a compact Lie group $G$, contains a direct summand isomorphic to $π_1(G)\oplusπ_1([G,G])$, where $[G,G]$ is the commutator subgroup of $G$. It follows from a similar statement for $E(2,G)$, the homotopy fiber of the canonical inclusion $B(2,G)\hookrightarrow BG$. As a consequence of our main result we obtain that if $E(2,G)$ is 2-connected, then $[G,G]$ is simply-connected. This last result completes how the higher connectivity of $E(2,G)$ resembles the higher connectivity of $[G,G]$ fo…
The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.
Link Homotopy in S^n x R^{m-n} and Higher Order mu-invariants
在[10]中，我们给出了到流形M上的一个合适的连接映射f，构造了连接同伦不变量kappa（f）和mu（f）.本文详细研究了M=S^n x R^{m-n}的情形.这里mu（f）是整个序列mu^（s）（f）的起始项，s = 0，1，.，更高的μ-不变量，它们一起捕获kappa（f）中包含的所有信息。我们讨论了这些新的不变量的几何意义。在几个实例中，我们获得了完整的分类结果。我们的方法的一个核心组成部分是同伦理论的楔形球。
For $r\geq 1$, the $r$-independence complex of a graph $G$ is a simplicial complex whose faces are subset $I \subseteq V(G)$ such that each component of the induced subgraph $G[I]$ has at most $r$ vertices. In this article, we determine the homotopy type of $r$-independence complexes of certain families of graphs including complete $s$-partite graphs, fully whiskered graphs, cycle graphs and perfect $m$-ary trees. In each case, these complexes are either homotopic to a wedge of equi-dimensional spheres or are contractible. We also give a closed form formula for their homotopy types.
We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_\infty$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.
This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $n\geq 2$. Second we exhibit for each $n\geq 1$ a natural homomorphism $ψ:A_n(G)\to \mathcal{H}_n(G)$, where $\mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $\mathcal{H}_n(G)$ is nontrivial and $ψ$ is surjective. It follows that for each $n\geq 1$ there are graphs $G$ for which $…
For every $\infty$-category $\mathscr{C}$, there is a homotopy $n$-category $\mathrm{h}_n \mathscr{C}$ and a canonical functor $γ_n \colon \mathscr{C} \to \mathrm{h}_n \mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Using homotopy $n$-categories, we introduce the notion of an $n$-derivator and study the main examples arising from $\infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $\infty$-derivators and prov…
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an $n$-type can be delooped $n+2$ times, then it is an infinite loop type. Most of the r…
