Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $π_1(M)$ is: (1)~$\mathbb{Z}^{*m}$, (2)~$\mathbb{Z}/p^r\mathbb{Z}$, or (3)~$\mathbb{Z}^{*m}*(*^n_{j=1}\mathbb{Z}/p_j^{r_j}\mathbb{Z})$, where $p$ and the $p_j$'s are odd primes.
We determine the number of distinct fibre homotopy types for the gauge groups of principal $Sp(2)$-bundles over a closed, simply-connected four-manifold.
Both the gauge groups and $5$-manifolds are important in physics and mathematics. In this paper, we combine them together to study the homotopy aspects of gauge groups over $5$-manifolds. For principal bundles over non-simply connected oriented closed $5$-manifolds of certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy gro…
In this paper we recover Bousfield's computation of $ν_1$-periodic homotopy groups of simply connected, finite $H$-spaces from \cite{Bou99} using the techniques of Goodwillie calculus. This is done through first computing André-Quillen cohomology over the monad $\mathbb{T}$ that encodes the power operations of complex $K$-theory. Then lifting this computation to computing $K$-theory of topological André-Quillen cohomology, and then using results of Behrens and Rezk relating it back to the Bousfield-Kuhn functor. The fact that we recovers the result of Bousfield allows us to conclude $ν_1$-peri…
In this paper, we study the homotopy groups of a shrinking wedge $X$ of a sequence $\{X_j\}$ of non-simply connected CW-complexes. Using a combination of generalized covering space theory and shape theory, we construct a canonical homomorphism $$Θ:π_n(X)\to\prod_{j\in\mathbb{N}}\bigoplus_{π_1(X)/π_1(X_j)}π_n(X_j),$$ characterize its image, and prove that $Θ$ is injective whenever each universal cover $\widetilde{X}_j$ is $(n-1)$-connected. These results (1) provide a characterization of the $n$-th homotopy group of the shrinking wedge of copies of $\mathbb{RP}^n$, (2) provide a characterizatio…
Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of 2 points in $M$, $F(M,2)$. When $M$ is even dimensional, we prove that the rational homotopy type of $F(M,2)$ depends only on the rational homotopy type of $M$. When the dimension of $M$ is odd, for every $x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $C(x)$. We prove that for some $x \in H^{n-2} (M; \mathbb{Q})$, $C(x)$ encodes completely the rational homotopy type of $F(M,2)$. For some class of manifolds, we show that we can…
Let $R$ be a principal ideal domain (PID). For a simply connected CW-complex $X$ of dimension $n$, let $Y$ be a space obtained by attaching cells of dimension $q$ to $X$, $q>n$, and let $A(Y)$ denote an Adams-Hilton model of $Y$. If $\mathcal E(A(Y))$ denotes the group of homotopy self-equivalences of $A(Y)$ and $\mathcal E_{*}(A(Y))$ its subgroup of the elements inducing the identity on $H_{*}( Y,R)$, then we construct two short exact sequences: $$\underset{i}{\oplus}\,H_{q}(ΩX,R)\rightarrowtail \mathcal{E}(A(Y))\overset{}{ \twoheadrightarrow}Γ^{q}_{n}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,…
We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.
The (co)homology self-closeness number of a simply-connected based CW-complexes $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions$\leq k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology self-closeness numbers of spaces in certain cofibrations, define the mod $p$ (co)homology self-closeness number of simply-connected $p$-local spaces with finitely generated homologies and stud…
We prove that completion at a set of primes preserves homotopy fibre squares of connected nilpotent spaces. As a consequence, we deduce the Hasse fracture square associated to a connected nilpotent space. Along the way, we give a quick proof of the well-known result that if the base of a fibre sequence has the homotopy type of a CW complex, then the total space has the homotopy type of a CW complex iff each fibre does.
In this paper we give a formula for the homotopy groups of $(n-1)$-connected $2n$-manifolds as a direct sum of homotopy groups of spheres in the case the $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore.
We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with well-behaved skeleta and some Poincaré duality features.
We first show that simply connected co-$H$-spaces and connected $H$-spaces can be uniquely decomposed into prime factors in the homotopy category of pointed $p$-local spaces of finite type, which is used to develop a $p$-local version of Gray's correspondence between homotopy types of prime co-$H$-spaces and homotopy types of prime $H$-spaces, and the split fibration which connects them as well. Further, we use the unique decomposition theorem to study the homotopy rigidity problem for classic functors. Among others, we prove that $ΣΩ$ and $Ω$ are homotopy rigid on simply connected $p$-local c…
For a field $\K$ of characteristic zero, we introduce a cohomologically symplectic poset structure ${\mathcal P}_{\K}(X)$ on a simply connected space $X$ from the viewpoint of $\K$-homotopy theory. It is given by the poset of inclusions of subgroups preserving c-symplectic structures in the group ${\mathcal E}(X_{\K})$ of $\K$-homotopy classes of $\K$-homotopy self-equivalences of $X$, which is defined by the $\K$-Sullivan model of $X$. We observe that the height of the Hasse diagram of ${\mathcal P}_{\K}(X)$ added by 1, denoted by c-s-${\rm depth}_{\K}(X)$, is finite and often depends on the …
Methods are developed to relate the action of a principal fibration to relative Whitehead products in order to determine the homotopy type of certain spaces. The methods are applied to thoroughly analyze the homotopy type of the based loops on certain cell attachments. Key examples are (n-1)-connected Poincare Duality complexes of dimension 2n or 2n+1 with minor cohomological conditions.
We develop a homotopy theory of $L_\infty$ algebras based on the Lawrence-Sullivan construction, a complete differential graded Lie algebra which, as we show, satisfies the necessary properties to become the right cylinder in this category. As a result, we obtain a general procedure to algebraically model the rational homotopy type of non-connected spaces.
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 …
For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M)$. Moreover, we show that these $p$-local homotopy groups can be expressed as direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work.
Crossed modules are known to be a model of pointed connected homotopy 2-types; formally, the homotopy category of crossed modules is equivalent to the category of pointed connected homotopy 2-types. In forming the homotopy category of crossed modules, one must resort to computing derived morphisms using non-constructive topological methods, but Behrang Noohi was able to find an algebraic model of these derived morphisms called butterflies. The result is a completely algebraic model of pointed connected homotopy 2-types. Reduced crossed complexes are a generalization of crossed modules and mode…
We introduce a rational homotopy invariant P of a topological space, which is a quintic tensor on the cohomology. For n > 1, formality of a closed (n-1)-connected manifold of dimension up to 5n-2 is equivalent to vanishing of P and the Bianchi-Massey tensor introduced by Crowley and the second author arXiv:1505.04184. We show also that elements of the group of closed simply-connected spin 8-manifolds with the cohomology of an r-fold connected sum of S^2 x S^6 are determined up to torsion by the value of P.
The self-closeness number $N\mathcal{E}(X)$ of a space $X$ is the least integer $k$ such that any self-map is a homotopy equivalence whenever it is an isomorphism in the $n$-th homotopy group for each $n\le k$. We discuss the self-closeness numbers of certain non-simply-connected $X$ in this paper. As a result, we give conditions for $X$ such that $N\mathcal{E}(X)=N\mathcal{E}(\widetilde{X})$, where $\widetilde{X}$ is the universal covering space of $X$.
In this paper, we study the stable homotopy types of $\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.
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.
Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. This is applied to classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.
For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.
This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it classifies the homotopy types of CW-complexes.
Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_\infty \colon UE \to \mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ ω\colon \mathrm{aut}_1(\mathrm{Baut}_1(X_{\mathbb Q})) \to \mathrm{Baut}_1(X_{\mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_\infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $\mathrm{Baut}_1(X_{\mathbb Q})$ as a consequence. We also prove tha…
Connected components of $\Map(S^4,B\SU(2))$ are the classifying spaces of gauge groups of principal $\SU(2)$-bundles over $S^4$. Tsukuda [Tsu01] has investigated the homotopy types of connected components of $\Map(S^4,B\SU(2))$. But unfortunately, the proof of Lemma 2.4 in [Tsu01] is not correct for $p=2$. In this paper, we give a complete proof. Moreover, we investigate the further divisibility of $ε_i$ defined in [Tsu01]. In [Tsu], it is shown that divisibility of $ε_i$ have some information about $A_i$-equivalence types of the gauge groups.
Let $M$ be a closed, oriented, simply connected 6-manifold. After localization away from 2, we give a homotopy decomposition of $ΣM$ in terms of spheres, Moore spaces and other recognizable spaces. As applications we calculate generalized cohomology groups of $M$ and determine the homotopy types of gauge groups of certain bundles over $M$.
Let $X$ be a \text{\rm{2}}-connected and \text{\rm{6}}-dimensional CW-complex $X$ such that $H_{3}(X)\otimes\Z_2=0$. This paper aims to describe the group $\E(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $\E_{*}(X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES$(X)$, we define the group $Γ\mathcal{S}(X)$ of $Γ$-automorphisms of WES$(X)$ and we prove that $\E(X)/\E_*(X)\cong Γ\mathcal{S}(X).$
We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology. This includes a recent generalization of Adams' cobar-construction to the non-simply connected case, and a new model for the homotopy theory of connected topological spaces using an infinity category of discrete monoids.
We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected space X. These are described in terms of simplicial resolutions of successive approximations (L^k,α} to the Quillen DGL model for X. The operations lie in suitable cohomology groups H^*{L^{(k,α)}; π_* X\otimes Q} of these DGLs. To facilitate the recovery of an integral version of the operations from the rational description, we also define a differential graded non-associative algebra model for rational spaces.
For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$.
For a simply connected rationally elliptic CW-complex $X$, we show that the cohomology and the homotopy Euler-Poincaré characteristics are related to two new numerical invariants namely $η_{X}$ and $ρ_{X}$ which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of $X$.
We study the asymptotic behavior of the homotopy groups of simply connected finite $p$-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the functorial decomposition of loop suspension, we prove that the suspended finite complexes are locally hyperbolic if suitable but accessible information of the homotopy groups is known. In particular, we prove that Moore spaces are locally hyperbolic, and other candidates are also given.
This paper extends the results of "Operads and Algebraic Homotopy" in giving algebraic invariants for the stable homotopy type of a pointed simply-connected simplicial set.
This paper proves that the homotopy type of a pointed, simply-connected, 2-reduced simplicial set is determined by the chain-complex augmented by functorial diagonal and higher diagonal maps (a simple generalization of the ones used to define Steenrod operations). The treatment of this problem is completely self-contained, and includes material that simplifies, extends, and corrects material from the authors AMS Memoir, "Iterating the cobar construction".
In this paper, we show that for a simply connected CW complex $Y$ with $H^{*}(Y;\mathbb{Q})$ of finite dimension, if $H^{*}(Y;\mathbb{Q})$ is concentrated in degrees $\leq 3$, then the rationalization $Y_\mathbb{Q}$ is formal. As an application, we show that the spatial realizations of simply connected Sullivan algebras with rational homology concentrated in degrees $\leq 3$ are formal.
Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $\mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $k\in\mathbb{Z}$. It is known that the homotopy type of $\mathcal{G}_k(M)$ is determined by the homotopy type of $\mathcal{G}_k(\mathbb{CP}^2)$. In this paper we investigate properties of $\mathcal{G}_k(\mathbb{CP}^2)$ when $G = SU(n)$ that partly classify the homotopy types of the gauge groups.
In this paper we study the rational homotopy of the space of immersions, $Imm\left(M,N\right)$, of a manifold $M$ of dimension $m\geq 0$ into a manifold $N$ of dimension $m+k$, with $k\geq 2$. In the special case when $N=\mathbb{R}^{m+k}$ and $k$ is odd we prove that each connected component of $Imm\left(M,\mathbb{R}^{m+k}\right)$ has the rational homotopy type of product of Eilenberg Mac Lane space. We give an explicit description of each connected component and prove that it only depends on $m$, $k$ and the rational Betti numbers of $M$. For a more general manifold $N$, we prove that the pat…
In this paper we study the relationship between the homology and homotopy of a space at infinity and at its boundary. Firstly, we prove that if a locally connected, connected, $δ$-hyperbolic space that is acted upon geometrically by a group has trivial homotopy at infinity then the first Čech homotopy group is trivial. Secondly, we prove that if a hyperbolic group on a finite field has trivial $i^{th}$ homology at infinity then the boundary of the group has trivial $i^{th}$ Steenrod homology. This result turns out to be important in addressing an open problem related to Cannon's conjecture.
We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of Church, and that their derived dual integral homotopy groups are finitely-generated as FI-modules in the sense of Church-Ellenberg-Farb. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-FI-space to properties of its dual homotopy groups. We also discuss several other applications.
In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type
This paper proves that the functor $C(*)$ that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, $C(*)$ is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies …
The purpose of this text is the study of the class of homotopy types which are modelized by strict \infty-groupoids. We show that the homotopy category of simply connected \infty-groupoids is equivalent to the derived category in homological degree greater or equal to 2 of abelian groups. We deduce that the simply connected homotopy types modelized by strict \infty-groupoids are precisely the products of Eilenberg-Mac Lane spaces. We also briefly study 3-categories with weak inverses. We finish by two questions about the problem suggested by the title of this text.
We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $π_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $ΩF$ have values in homotopy n-nilpotent groups.
Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set $X^{hG}$, and the natural injection $k\colon X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, tha…
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l o…
