For the n-dimensional Hawaiian earring ${ℍ}_n,$ n ≥ 2, ${\pi }_n({ℍ}_n,o)\simeq {\mathbb{Z}}^{\omega }$ and ${\pi }_i({ℍ}_n,o)$ is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then $H_n\left(X\vee Y\right)\simeq H_n\left(X\right)\oplus H_n\left(Y\right)\oplus H_n\left(CX\vee CY\right)$ for n ≥ 1.

Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors ${\Omega }_{\lambda }\left(R\right)$ and ℓ(R) together with algebra homomorphisms $e:{\Omega }_{\lambda }\left(R\right)\to H*\left(\Lambda X\right)$ and $\psi :\ell \left(R\right)\to H*\left(E{S}^1{\times }_{S^1}\Lambda X\right)$. When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a $$𝔤{𝔩}_n$$ partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.

We classify the indecomposable injective E(n)⁎E(n)-comodules, where E(n) is the Johnson-Wilson homology theory. They are suspensions of the $J_{n,r}=E\left(n\right)⁎\left(M_{r}E\left(r\right)\right)$, where 0 ≤ r ≤ n, with the endomorphism ring of $J_{n,r}$ being $\widehat{E\left(r\right)}*\widehat{E\left(r\right)}$, where $\widehat{E\left(r\right)}$ denotes the completion of E(r).

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of ${\pi }_2$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional...

Using the Berline-Vergne integration formula for equivariant cohomology for torus actions, we prove that integrals over Grassmannians (classical, Lagrangian or orthogonal ones) of characteristic classes of the tautological bundle can be expressed as iterated residues at infinity of some holomorphic functions of several variables. The results obtained for these cases can be expressed as special cases of one formula involving the Weyl group action on the characters of the natural representation of...

We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.