Volume 305, Nº 1 (2019)

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment–angle complex \({{\cal Z}_{\cal K}}\). Namely, we say that a simplicial complex \({\cal K}\) realises an iterated higher Whitehead product w if w is a nontrivial element of \({\pi _*}\left( {{{\cal Z}_{\cal K}}} \right)\). The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product w we describe a simplicial complex ∂Δ_w that realises w. Furthermore, for a particular form of brackets inside w, we prove that ∂Δ_w is the smallest complex that realises w. We also give a combinatorial criterion for the nontriviality of the product w. In the proof of nontriviality we use the Hurewicz image of w in the cellular chains of \({{\cal Z}_{\cal K}}\) and the description of the cohomology product of \({{\cal Z}_{\cal K}}\). The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complexes for the face coalgebra of \({\cal K}\) to describe the canonical cycles corresponding to iterated higher Whitehead products w. This gives another criterion for realisability of w.

An elliptic function of level N determines an elliptic genus of level N as a Hirzebruch genus. It is known that any elliptic function of level N is a specialization of the Krichever function that determines the Krichever genus. The Krichever function is the exponential of the universal Buchstaber formal group. In this work we give a specialization of the Buchstaber formal group such that this specialization determines formal groups corresponding to elliptic genera of level N. Namely, an elliptic function of level N is the exponential of a formal group of the form F(u, v) = (u²A(v) − v²A(u))/(uB(v) − vB(u)), where A(u), B(u) ∈ ℂ[[u]] are power series with complex coefficients such that A(0) = B(0) = 1, A″(0) = B′(0) = 0, and for m = [(N − 2)/2] and n = [(N − 1)/2] there exist parameters (a₁, …, a_m, b₁, …, b_n) for which the relation \(\prod\nolimits_{j=1}^{n-1}\left(B(u)+b_{j} u\right)^{2} \cdot\left(B(u)+b_{n} u\right)^{N-2 n}=A(u)^{2} \prod\nolimits_{k=1}^{m-1}\left(A(u)+a_{k} u^{2}\right)^{2} \cdot\left(A(u)+a_{m} u^{2}\right)^{N-1-2 m}\) holds. For the universal formal group of this form, the exponential is an elliptic function of level at most N. This statement is a generalization to the case N > 2 of the well-known result that the elliptic function of level 2 determining the elliptic Ochanine–Witten genus is the exponential of a universal formal group of the form F(u, v) = (u² − v²)/(uB(v) − vB(u)), where B(u) ∈ ℂ[[u]], B(0) = 1, and B′(0) = 0. We prove this statement for N = 3, 4, 5, 6. We also prove that the elliptic function of level 7 is the exponential of a formal group of this form. Universal formal groups that correspond to elliptic genera of levels N = 5, 6, 7 are obtained in this work for the first time.

We consider three uniqueness theorems: one from the theory of meromorphic functions, another from asymptotic combinatorics, and the third concerns representations of the infinite symmetric group. The first theorem establishes the uniqueness of the function exp z in a class of entire functions. The second is about the uniqueness of a random monotone nonde-generate numbering of the two-dimensional lattice ℤ₊², or of a nondegenerate central measure on the space of infinite Young tableaux. And the third theorem establishes the uniqueness of a representation of the infinite symmetric group \(\mathfrak{S}_{\mathbb{N}}\) whose restrictions to finite subgroups have vanishingly few invariant vectors. However, in fact all the three theorems are, up to a nontrivial rephrasing of conditions from one area of mathematics in terms of another area, the same theorem! Up to now, researchers working in each of these areas have not been aware of this equivalence. The parallelism of these uniqueness theorems on the one hand and the difference of their proofs on the other call for a deeper analysis of the nature of uniqueness and suggest transferring the methods of proof between the areas. More exactly, each of these theorems establishes the uniqueness of the so-called Plancherel measure, which is the main object of our paper. In particular, we show that this notion is general for all locally finite groups.

We recall the construction and survey the properties of the Yamada polynomial of spatial graphs and present formulas for the Yamada polynomial of some classes of graphs. Then we construct an infinite family of spatial graphs for which the roots of the Yamada polynomials are dense in the complex plane.

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply the existence of a higher Massey product in the cohomology of a moment-angle complex \({{\cal Z}_K}\), and this product contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family \({\cal F}\) of polyhedral products being smooth closed manifolds such that for any l, r ≥ 2 there exists an l-connected manifold \(M \in {\cal F}\) with a nontrivial strictly defined r-fold Massey product in H*(M). As an application to homological algebra, we determine a wide class of triangulated spheres K such that a nontrivial higher Massey product of any order may exist in the Koszul homology of their Stanley–Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph Γ to provide a (rationally) formal generalized moment-angle manifold \(\mathcal{Z}_{P}^{J}=\left(\underline{D}^{2 j_{i}}, \underline{S}^{2 j_{i}-1}\right)^{\partial P^{*}}\)J = (j₁,…,j_m), over a graph-associahedron P = P_Γ, and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

The work is devoted to the study of one-point commuting difference operators of rank 2. In the case of hyperelliptic spectral curves, we obtain equations equivalent to the Krichever–Novikov equations for the discrete dynamics of the Tyurin parameters. Using these equations, we construct examples of operators corresponding to hyperelliptic spectral curves of arbitrary genus.

An Alexander self-dual complex gives rise to a compactification of \({{\cal M}_{0,n}}\), called an ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the configuration spaces of flexible polygons. We present an explicit description of the Chow rings of ASD compactifications. We study the analogs of Kontsevich’s tautological bundles, compute their Chern classes, compute top intersections of the Chern classes, and derive a recursion for the intersection numbers.

For a p-toral group G, we answer the question which compact (respectively, open) smooth manifolds M can be diffeomorphic to the fixed point sets of smooth actions of G on compact (respectively, open) smooth manifolds E of the homotopy type of a finite ℤ-acyclic CW complex admitting a cellular map of period p, with exactly one fixed point. In the case where the CW complex is contractible, E can be chosen to be a disk (respectively, Euclidean space).

We determine the number of distinct homotopy types for the gauge groups of principal Sp(2)-bundles over a closed simply connected four-manifold.

Regular semisimple Hessenberg varieties are subvarieties of the flag variety Flag(ℂⁿ) arising naturally at the intersection of geometry, representation theory, and combinatorics. Recent results of Abe, Horiguchi, Masuda, Murai, and Sato as well as of Abe, DeDieu, Galetto, and Harada relate the volume polynomials of regular semisimple Hessenberg varieties to the volume polynomial of the Gelfand–Zetlin polytope GZ(λ) for λ = (λ₁, λ₂, …, λ_n). In the main results of this paper we use and generalize tools developed by Anderson and Tymoczko, by Kiritchenko, Smirnov, and Timorin, and by Postnikov in order to derive an explicit formula for the volume polynomials of regular semisimple Hessenberg varieties in terms of the volumes of certain faces of the Gelfand–Zetlin polytope, and also exhibit a manifestly positive, combinatorial formula for their coefficients with respect to the basis of monomials in the α_i:= λ_i − λ_i+1. In addition, motivated by these considerations, we carefully analyze the special case of the permutohedral variety, which is also known as the toric variety associated to Weyl chambers. In this case, we obtain an explicit decomposition of the permutohedron (the moment map image of the permutohedral variety) into combinatorial (n − 1)-cubes, and also give a geometric interpretation of this decomposition by expressing the cohomology class of the permutohedral variety in Flag(ℂⁿ) as a sum of the cohomology classes of a certain set of Richardson varieties.