The description for this book, Compositional Methods in Homotopy Groups of Spheres. (AM-49), will be forthcoming.

Composition methods in homotopy groups of spheres

The relations M (κ, λ, µ) → B [resp. B(σ)] meaning that if A ⊂ [κ] λ with |A| = κ is µ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let be any regular cardinal with a supercompact [resp. 2-huge] cardinal above. Then there is a-closed forcing P such that, in V P , we have both GCH and M ((++1) , + ,) B [resp. M ((++1) , λ,) B(+) for all λ ≤ (++1) ]. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz]. 1. Introduction. The aim of this paper is to show that, assuming the existence of certain large cardinals, the results of [EH] are sharp. Let us recall these results, and first their terminology. If µ ≤ λ ≤ κ and σ are infinite cardinals then M (κ, λ, µ) → B(σ) [resp. M (κ, λ, µ) → B] abbreviates the following statement: Whenever A ⊂ [κ]

Fundamenta Mathematicae私抄 : 退任の辞に代えて

Classically, a spin structure on the loop space of a manifold is a lift of the structure group of the looped frame bundle from the loop group to its universal central extension. Heuristically, the loop space of a manifold is spin if and only if the manifold itself is a string manifold, against which it is well-known that only the if-part is true in general. In this article we develop a new version of spin structures on loop spaces that exists if and only if the manifold is string, as desired. This new version consists of a classical spin structure plus a certain fusion product related to loops of frames in the manifold. We use the lifting gerbe theory of Carey-Murray, recent results of Stolz-Teichner on loop spaces, and some own results about string geometry and Brylinski-McLaughlin transgression.

/pdf/spin-structures-on-loop-spaces-that-characterize-string-cbuww58d5a.pdf

Spin structures on loop spaces that characterize string manifolds

Let $\pi: P\to B$ be a locally trivial fiber bundle over a connected CW complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then $\pi$ is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms $\Symp(M,\om)$, and is called Hamiltonian if this group may be reduced to the group $\Ham(M,\om)$ of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases $B$ (for example when $B$ is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of $P$ is the tensor product of the cohomology of $B$ with that of $M$. As a consequence the natural action of the rational homology $H_k(\Ham(M))$ on $H_*(M)$ is trivial for all $M$ and all $k > 0$. 
Added: The erratum makes a small change to Theorem 1.1 concerning the characterization of Hamiltonian bundles.

Symplectic Structures on Fiber Bundles

We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y) We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX We consider these topics in both ordinary homotopy theory as well as after localization In the latter case, we discuss algebraic models for the localization of function spaces and their applications

The homotopy theory of function spaces: a survey

Let $B{ aut}_1X$ be the Dold-Lashof classifying space of orientable fibrations with fiber $X$. For a rationally weakly trivial map $f:X\to Y$, our strictly induced map $a_f: (Baut_1X)_0\to (Baut_1Y)_0$ induces a natural map from a $X_0$-fibration to a $Y_0$-fibration. It is given by a map between the differential graded Lie algebras of derivations of Sullivan models. We note some conditions that the map $a_f$ admits a section and note some relations with the Halperin conjecture. Furthermore we give the obstruction class for a lifting of a classifying map $h: B\to (Baut_1Y)_0$ and apply it for liftings of $G$-actions on $Y$ for a compact connected Lie group $G$ as the case of $B=BG$ and evaluating of rational toral ranks as $r_0(Y)\leq r_0(X)$.

An induced map between rationalized classifying spaces for fibrations

A poset-stratified space is a pair $(S, S \xrightarrow \pi P)$ of a topological space $S$ and a continuous map $\pi: S \to P$ with a poset $P$ considered as a topological space with its associated Alexandroff topology. In this paper we show that one can impose such a poset-stratified space structure on the homotopy set $[X, Y]$ of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set $[X, Y]$ as an object of the category of poset-stratified spaces. The order we consider is related to the notion of \emph{dependence of maps} (by Karol Borsuk). Furthermore via homology and cohomology the homotopy set $[X,Y]$ can have other poset-stratified space structures. In the cohomology case, we get some results which are equivalent to the notion of \emph{dependence of cohomology classes} (by Rene Thom) and we can show that the set of isomorphism classes of complex vector bundles can be captured as a poset-stratified space via the poset of the subrings consisting of all the characteristic classes. We also show that some invariants such as Gottlieb groups and Lusternik--Schnirelmann category of a map give poset-stratified space structures to the homotopy set $[X,Y]$

Poset-stratified space structures of homotopy sets.

For a map 
$$f:X\rightarrow Y$$

, there is the relative model 
$$M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)\simeq M(X)$$

 by Sullivan model theory (Felix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let 
$$\mathrm{Baut}_1X$$

 be the Dold–Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations 
$$\mathrm{Der}M(X)$$

 of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction 
$$b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) $$

 is a DGL-map and the related topics.

/pdf/on-a-dgl-map-between-derivations-of-sullivan-minimal-models-3a7l6ilhpy.pdf

On a DGL-map between derivations of Sullivan minimal models

We define the fibre-restricted Gottlieb group with respect to a fibration $\xi :X\to E\to Y$ in CW complexes. It is a subgroup of the Gottlieb group of $X$. When $X$ and $E$ are finite simply connected, its rationalized model is given by the arguments of derivations of Sullivan models based on Felix, Lupton and Smith \cite{FLS}. We consider the realization problem of groups in a Gottlieb group as fibre-restricted Gottlieb groups in rational homotoy theory. Especially we define an invariant named as (Gottlieb) depth of $X$ over $Y$. In particular, when $Y=BS^1$, it is related to the rational toral rank of $X$.

A rational realization problem in Gottlieb group

pdf/poset-stratified-space-structures-of-homotopy-sets-5cxhuhqccb.pdf

Poset-stratified space structures of homotopy sets

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