The evaluation subgroup of a fibre inclusion

A rational realization problem in Gottlieb group

We define the fibre-restricted Gottlieb group with respect to a fibration $ξ:X\to E\to Y$ 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 Félix, 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…

The rational homotopy type of the space of self-equivalences of a fibration

Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).

纤维化轨道流形解消及其在$\mathrm{G}_2$-模空间上的应用

Fibrewise Orbifold Resolutions with Applications to $\mathrm{G}_2$-Moduli Spaces

通过扭曲爆破族解消定制轨道流形的奇点，构造流形丛，并利用实同伦理论证明这些丛在$\pi_2(B\mathrm{hAut}(M)_0)$中生成自由子群，进而证明广义Kummer构造得到的无挠$\mathrm{G}_2$结构模空间的第二同伦群包含正秩自由子群。

Rationalized Evaluation Subgroups of a Map and the Rationalized G-Sequence

Let f: X -> Y be a based map of simply connected spaces. The corresponding evaluation map w: map(X,Y;f) -> Y induces a homomorphism of homotopy groups whose image in pi_n(Y) is called the nth evaluation subgroup of f. The nth Gottlieb group of X occurs as the special case in which Y = X and f = 1_X. We identify the homomorphism induced on rational homotopy groups by this evaluation map, in terms of a map of complexes of derivations constructed using Sullivan minimal models. Our identification allows for the characterization of the rationalization of the nth evaluation subgroup of f. It also al…

The rational cohomology groups of the classifying spaces of Kac-Moody groups

In this paper, we compute the rational cohomology groups of the classifying space of a simply connected Kac-Moody group of infinite type. The fundamental principle is "from finite to infinite". That is, for a Kac-Moody group G(A) of infinite type, the input data for computation are the rational cohomology of classifying spaces of parabolic subgroups of G(A)(which are of finite type), and the homomorphisms induced by inclusions of these subgroups. In some special cases, we can further determine the cohomology rings. Our method also applies to study the mod p cohomology of the classifying spaces…

关于多面体乘积的Félix-Tanré有理模型

On Félix-Tanré rational models for polyhedral products

本文研究纤维包含的多面体乘积的Félix-Tanré有理模型，探究李群对多面体乘积的有理模型与有理同伦群，并证明环面流形相关偏商的三个等价条件。

The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (π\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was late…

Alternative stable homotopy classification of p-completed classifying spaces

We give an alternative to the stable classification of p-completed homotopy types of classifying spaces of finite groups offered by Martino-Priddy. For a finite group G with Sylow subgroup S, we regard the stable p-completed classifying space of G as an object under the stable p-completed classifying space of S via the canonical inclusion map. Thus we get a classification in terms of induced fusion systems. Applying Oliver's solution to the Martino-Priddy conjecture, we obtain the surprising result that the unstable p-completed homotopy type of BG is determined by the stable p-completed inclus…