We continue the analysis, started by Abreu, McDuff and Anjos, of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.
The symplectomorphism group of a 2-dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of the symplectic 4-manifold (M, omega) into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface Sigma which is Poincare dual to a multiple of the form omega. We show that then one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L, Sigma). This allows us to compute the homoto…
We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp \TM and Diff \TM of its one point blow up \TM. There are three main arguments. The first shows that for any oriented M the natural map from pi_1(M) to pi_0(Diff \TM) is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group pi_1(Diff \TM) that persist into the space of self homotopy equivalences of \TM. Since it uses purely homological arguments, it applies to …
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.
We analyse the homotopy types of gauge groups of principal U(n)-bundles associated to pseudo Real vector bundles in the sense of Atiyah. We provide satisfactory homotopy decompositions of these gauge groups into factors in which the homotopy groups are well known. Therefore, we substantially build upon the low dimensional homotopy groups as provided in a paper by I. Biswas, J. Huisman, and J. Hurtubise.
We give $p$-local homotopy decompositions of the suspensions of real toric spaces for odd primes $p$. Our decomposition is compatible with the one given by Bahri, Bendersky, Cohen, and Gitler for the suspension of the corresponding real moment-angle complex, or more generally, the polyhedral product. As an application, we obtain a stable rigidity property for real toric spaces.
Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Felix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Las…
Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the $p$-exponent of $W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and $0<m\leq (p-1)(p-2)$ are computed as well.
We use Richter's $2$-primary proof of Gray's conjecture to give a homotopy decomposition of the fibre $Ω^3S^{17}\{2\}$ of the $H$-space squaring map on the triple loop space of the $17$-sphere. This induces a splitting of the mod-$2$ homotopy groups $π_\ast(S^{17}; \mathbb{Z}/2\mathbb{Z})$ in terms of the integral homotopy groups of the fibre of the double suspension $E^2:S^{2n-1} \to Ω^2S^{2n+1}$ and refines a result of Cohen and Selick, who gave similar decompositions for $S^5$ and $S^9$. We relate these decompositions to various Whitehead products in the homotopy groups of mod-$2$ Moore spa…
Given a space $X$ and a simplicial complex $K$ with $m$-vertices, the arrangement of partially diagonal subspaces of $X^m$, called the dragonal arrangement, is defined. We decompose the suspension of the diagonal arrangement when $2(dim K + 1) < m$, which generalizes the result of Labassi. As a corollary, we calculate the Euler characteristic of the complement when $X$ is a closed connected manifold.
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.
We provide a family of spaces localized at 2, whose stable homotopy groups are summands of their unstable homotopy groups. Application to mod 2 Moore spaces are given.
We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.
We describe Bott towers as sequences of toric manifolds M^k, and identify the omniorientations which correspond to their original construction as toric varieties. We show that the suspension of M^k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's recent analysis of the Adams Spectral Sequence. By way of application we inves…
We give p-local homotopy decompositions of the loop spaces of compact, simply-connected symmetric spaces for quasi-regular primes. The factors are spheres, sphere bundles over spheres, and their loop spaces. As an application, upper bounds for the homotopy exponents are determined.
We study the triangulated subcategories of compact objects in stable homotopy categories such as the homotopy category of spectra, the derived categories of rings, and the stable module categories of Hopf algebras. In the first part of this thesis we use a K-theory recipe of Thomason to classify these subcategories. This recipe when applied to the category of finite p-local spectra gives a refinement of the ``chromatic tower''. This refinement has some interesting consequences. In particular, it gives new evidence to a conjecture of Frank Adams that the thick subcategory C_2 can be generated b…
Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension $Σ^2M$ as wedge sums of some elementary $\mathbf{A}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of $Σ^2M$ to desuspend to that of $ΣM$.
A free homotopy decomposition of any continuous map from a compact Riemmanian manifold $\mathcal{M}$ to a compact Riemannian manifold $\mathcal{N}$ into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map in $W^{s,p} (\mathcal{M}, \mathcal{N})$, with $sp = m = \dim \mathcal{M}$. In particular, when the fundamental group $π_1 (\mathcal{N})$ acts trivially on the homotopy group $π_m (\ma…
The filtration $\operatorname{BGL}_{0}\subset\dots\subset\operatorname{BGL}_{n-1}\subset\operatorname{BGL}_{n}$ is split by motivic Becker-Gottlieb transfers in the motivic stable homotopy category over any scheme. This recovers results by Snaith on the splitting of $\operatorname{BGL}_{n}(\mathbb{C})$ in classical stable homotopy theory by passing to complex realizations. On the way, we extend motivic homotopy theory to smooth ind-schemes as bases and show how to construct the necessary fragment of the six operations and duality for this extension.
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.
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 $M$.
In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a \textit{Hodge decomposition} on $(X\otimes\mathbb{C})^{sch}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb{C}^{\times δ}$ on the object $(X\otimes…
Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition theorems for thick subcategories of various triangulated categories including the derived categories of rings, Noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. We also discuss some consequences of these decomposition results. In particular, it is shown that all these decompositions respect K-theory.
Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recov…
A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral product $Z_K(X,A)$ whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include $p$-local and rational versions of the theorem. We end with a group theoretic application.
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.
Fix an odd prime $p$ and let $X$ be the $p$-localization of a finite suspended $CW$-complex. Given certain conditions on the reduced mod-$p$ homology $\bar H_*(X;\zmodp)$ of $X$, we use a decomposition of $ΩΣX$ due to the second author and computations in modular representation theory to show there are arbitrarily large integers $i$ such that $ΩΣ^i X$ is a homotopy retract of $ΩΣX$. This implies the stable homotopy groups of $ΣX$ are in a certain sense retracts of the unstable homotopy groups, and by a result of Stanley, one can confirm the Moore conjecture for $ΣX$. Under additional assumptio…
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.
In this paper we prove that the quotient of any real or complex moment-angle complex by any closed subgroup in the naturally acting compact torus on it is equivariantly homotopy equivalent to the homotopy colimit of a certain toric diagram. For any quotient we prove an equivariant homeomorphism generalizing the well-known Davis-Januszkiewicz construction for quasitoric manifolds and small covers. We deduce formality of the corresponding Borel construction space under the natural assumption on the group action in the complex case leading to the new description of the equivariant cohomology for …
Let G be a finite group. We study the group of G-equivariant self-homotopy equivalences of product of G-spaces. For a product of n-spaces, we represent it as product of n-subgroups under the assumption of equivariant reducibility. Further we describe each factor as a split short exact sequence. Also, we obtain an another kind of factorisation, called $LU$ type decomposition, as product of two subgroups.
In "Homotopy decomposition of classifying spaces via elementary Abelian subgroups", Stephan Jackowski and James McClure show, for functors admitting a Mackey complement over categories holding a direct product, a general result on vanishing cohomology. We develop a framework leading to a general result on trivial homotopy which partially generalizes Jackowski and McClure's result in two different directions.
We prove two homotopy decomposition theorems for the loops on co-H-spaces, including a generalization of the Hilton-Milnor Theorem. These are applied to problems arising in algebra, representation theory, toric topology, and the study of quasi-symmetric functions.
It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (See Theorem 21.3 of \cite{Fuchs:abelian-group}). In this paper, conditions to split off rational parts in homotopy types from a given space are studied in terms of a variant of Hurewicz map, say $\barρ : [S_{\Q}^{n},X] \to H_n(X;\Z)$ and generalized Gottlieb groups. This yields decomposition theorems on rational homotopy types of Hopf spaces, $T$-spaces and Gottlieb spaces, which has been known in various situations, especially for spaces with finiteness conditions.
Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex Y. We give a selection of results about certain subgroups of E(Y). We establish a connection between the Gottlieb groups of Y and the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of Y, denoted by E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of a cone decomposition of Y. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of E(Y) consist…
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…
Let $S^{2n+1}\{p\}$ denote the homotopy fibre of the degree $p$ self map of $S^{2n+1}$. For primes $p \ge 5$, work of Selick shows that $S^{2n+1}\{p\}$ admits a nontrivial loop space decomposition if and only if $n=1$ or $p$. Indecomposability in all but these dimensions was obtained by showing that a nontrivial decomposition of $ΩS^{2n+1}\{p\}$ implies the existence of a $p$-primary Kervaire invariant one element of order $p$ in $π_{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for $ΩS^{2n+1}\{p\}$ is equivalent to the strong…
Let $Y$ be a pointed space and let $\mathcal E(Y^r)$ be the group of based self-equivalences of $Y^r$, $r\geq 2$. For $Y$ a homotopy commutative $H$-group we construct a subgroup $\mathcal E_{\mathrm{Mat}}(Y^r)$ of $\mathcal E(Y^r)$ which has a group structure isomorphic to either $GL_r(\mathbb Z)$, or $GL_r(\mathbb Z_d)$, $d\geq 2$. We classify principal bundles over connected sums of $q$-sphere bundles over $n$-spheres and use the group $\mathcal E_{\mathrm{Mat}}(Y^r)$ to obtain homotopy decompositions of their gauge groups. Using these decompositions we give an integral classification, up t…
In this paper, we determine the homotopy groups π_4(ΣK(A,1)) and π_5(ΣK(A,1)) for abelian groups A by using different facts and methods from group theory and homotopy theory: derived functors, the Carlsson simplicial construction, the Baues-Goerss spectral sequence, homotopy decompositions and the methods of algebraic K-theory. As the applications, we also determine π_i(ΣK(G,1)) with i=4,5 for some non-abelian groups G=Σ_3 and SL(Z), and π_4(ΣK(A_4,1)) for the 4-th alternating group A_4.
The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes we show that this splitting on the level of the bialgebra of chord d…
Two homotopy decompositions of supensions of spaces involving polyhedral products are given. The first decomposition is motivated by the decomposition of suspensions of polyhedral products by Bahri, Bendersky, Cohen, and Gitler, and is a generalization of the retractile argument of James. The second decomposition is on the union of an arrangement of subspaces called diagonal subspaces, and generalizes the result of Labbasi.
We show that if X is any proper complex variety, there is a weight decomposition on the real schematic homotopy type, in the form of an algebraic G_m-action. This extends to a real Hodge structure, in the form of a discrete C^*-action, such that C^* x X -> X^{sch} is real analytic. If the fundamental group is algebraically good, and the higher homotopy groups have finite rank, this gives bigraded decompositions on the complexified homotopy groups. For smooth proper varieties, the Hodge structure can be recovered from the cohomology ring with coefficients in the universal semisimple local syste…
The purpose of this paper is to generalise Sullivan's rational homotopy theory to non-nilpotent spaces, providing an alternative approach to defining Toen's schematic homotopy types over any field k of characteristic zero. New features include an explicit description of homotopy groups using the Maurer-Cartan equations, convergent spectral sequences comparing schematic homotopy groups with cohomology of the universal semisimple local system, and a generalisation of the Baues-Lemaire conjecture. For compact Kaehler manifolds, the schematic homotopy groups can be described explicitly in terms of…
We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors (such as the degree 3 component $SP^3$ of the symmetric algebra functor). As an illustration of this method, we retrieve in a purely algebraic manne…
Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $lσ+mρ\in{π_4(SO(4))}$, $l,m\in\mathbb Z$. 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: $SU(n)$ $(n\geq4)$, $Sp(n)$ $(n\geq2)$, $Spin(n)$ $(n\geq5)$, $F_4$, $E_6$, $E_7$, $E_8$. 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 recognisab…
The homotopy theory of gauge groups has received considerable attention in recent decades. In this work, we study the homotopy theory of gauge groups over some high dimensional manifolds. To be more specific, we study gauge groups of bundles over $(n-1)$-connected closed $2n$-manifolds, the classification of which was determined by Wall and Freedman in the combinatorial category. We also investigate the gauge groups of the total manifolds of sphere bundles based on the classical work of James and Whitehead. Furthermore, other types of $2n$-manifolds are also considered. In all the cases, we sh…
We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying ho…
