Jelena Grbić, Abigail Linton
We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.
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As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of …
Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.
Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the co…
Let us say that a map of arcwise connected topological spaces (having the homotopy type of CW-complexes) is a pseudo-homeomorphism if it induces an isomorphism of the first integer homology groups and an epimorphism of the second integer homology groups. We prove that any invariant of a topological space w.r.t. pseudo-homeomorphisms is an invariant of the fundamental group of this space. We also describe a necessary condition for the fundamental groups to be distinguished by such invariants. As an example we show that the invariant used in math.AG/9805056 to distinguish the fundamental groups …
We construct splitting varieties for triple Massey products. For a,b,c in F^* the triple Massey product < a,b,c> of the corresponding elements of H^1(F, mu_2) contains 0 if and only if there is x in F^* and y in F[\sqrt{a}, \sqrt{c}]^* such that b x^2 = N_{F[\sqrt{a}, \sqrt{c}]/F}(y), where N_{F[\sqrt{a}, \sqrt{c}]/F} denotes the norm, and F is a field of characteristic different from 2. These varieties satisfy the Hasse principle by a result of D.B. Lee and A.R. Wadsworth. This shows that triple Massey products for global fields of characteristic different from 2 always contain 0.
Non-realizability of a triple Massey product for the algebra $\mathbb{F}_2[a,b,c]/(ab,bc)$
该研究证明了特定代数的三重Massey乘积无法被实现，通过计算Hochschild上同调群揭示了其本质形式特性。
We show that, in general, there exist non-vanishing triple Massey products in the cohomology with finite field coefficients of a complex hypersurface complement. In contrast, the Massey products, triple and higher, in the rational cohomology of such a space are all known to vanish.
We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.
Closed (and simply-connected) manifolds whose dimensions are larger than 4 are central geometric objects in classical algebraic topology and differential topology. They have been classified via algebraic and abstract objects. On the other hand, It is difficult to understand them in geometric and constructive ways. In the present paper, we show such studies via explicit fold maps, higher dimensional versions of Morse functions. The author captured information of the topologies and the differentiable structures of closed (and simply-connected) manifolds which are not so complicated with respect …
The problem of existence of nontrivial Massey products in cohomology of a space is well-known in algebraic topology and homological algebra. A number of problems in complex geometry, symplectic geometry, and algebraic topology can be stated in terms of Massey products. One of such problems is to establish formality of smooth manifolds in rational homotopy theory. There have already been constructed a few classes of spaces with nontrivial triple Massey products in cohomology. Until now, very few examples of manifolds $M$ with nontrivial higher Massey products in $H^*(M)$ were known. In this wor…
In this survey, we discuss two research areas related to Massey's higher operations. The first direction is connected with the cohomology of Lie algebras and the theory of representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley-Reisner rings of simplicial complexes.
We give a correct statement and a complete proof of the criterion obtained by Grbić, Panov, Theriault and Wu for the face ring $\Bbbk[K]$ of a simplicial complex $K$ to be Golod over a field $\Bbbk$. (The original argument depended on the main result of a paper by Berglund and Jöllenbeck, which was shown to be false by Katthän.) We also construct an example of a minimally non-Golod complex $K$ such that the cohomology of the corresponding moment-angle complex $\mathcal Z_K$ has trivial cup product and a non-trivial triple Massey product.
On the behavior of Massey products under field extension
本文研究了交换微分分次代数上 Massey 乘积在域扩张下的行为，证明其整体消没性不保持不变，并揭示了三重与高阶乘积在代数闭域扩张下的不同表现。
Strong minimal model theorem and Massey products
本研究证明了比Kadeishvili定理更强的、同痕意义下唯一的极小模型存在性，揭示了该结果与三重及高阶Massey积的对应关系，阐明了二者间的关联。
Poincaré dualization and Massey products
本文研究了庞加莱对偶空间中马塞产品的性质，揭示了非零度映射下四重马塞产品可能为平凡的现象，并探讨了形式性与庞加莱对偶空间的关系。
Bigraded notions of formality and Aeppli-Bott-Chern-Massey products
本文定义复流形微分形式代数的双分次形式性，并研究其与高阶 Aeppli–Bott–Chern–Massey 乘积的关系，证明低次乘积可非平凡拉回到沿复子流形的爆破。
We construct a family of manifolds, one for each $n\geq 2$, having a nontrivial Massey $n$-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus action called moment-angle manifolds $\mathcal Z_P$, whose orbit spaces are simple $n$-dimensional polytopes $P$ obtained from a $n$-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes $P$ are flag nestohedra but not graph-associahedra. We compute some bigraded Betti numbers $β^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in terms of it…
We show that for any face $F$ of a simple polytope $P$ the canonical equivariant homeomorphisms $h_P:\,\mathcal Z_P\to\mathcal Z_{K_P}$ and $h_F:\,\mathcal Z_F\to\mathcal Z_{K_F}$ are linked in a pentagonal commutative diagram with the maps of moment-angle manifolds and moment-angle-complexes, induced by a face embedding $i_{F,P}:\,F\to P$ and a simplicial embedding $Φ_{F,P}:\,K_F\to K_{F,P}\to K_P$, where $K_{F,P}$ is the full subcomplex of $K_P$ on the same vertex set as $Φ_{F,P}(K_F)$. We introduce the explicit constructions of the maps $i_{F,P}$, $Φ_{F,P}$ and show that a polytope $P$ is f…
Motivated by the construction of Steenrod cup-$i$ products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential g…
Highly connected non-formal Milnor fibers via polyhedral products
通过结合Fernández de Bobadilla的实现定理和Grbić-Linton的系统Massey积构造，产生高度连接且非形式的Milnor纤维。
We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples are useful for theory of systols.
This paper starts with an observation that two infinite series of simplicial complexes, which a priori do not seem to have anything to do with each other, have the same homotopy type. One series consists of the complexes of directed forests on a double directed string, while the other one consists of Shapiro-Welker models for the spaces of hyperbolic polynomials with a triple root. We explain this coincidence in the more general context by finding an explicit homotopy equivalence between complexes of directed forests on a double directed tree, and doubly disconnecting complexes of a tree.
We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional mani…
We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because …
The rational homology of independence complexes of Kneser graphs on triples
该研究针对三元组上的Kneser图独立复形，在有理数域上肯定回答了Feng和Wang提出的同调非平凡维度问题，确定了其非零Betti数，通过多种同调工具完成了证明。
Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms …
A Generalization of Arrow's Impossibility Theorem Through Combinatorial Topology
本文通过组合拓扑的2-骨架给出阿罗定理的新证明，并在包含无限制域的极化与多样性域限制下推广该定理，展示了组合拓扑方法在高维域限制研究中的有效性。
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