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 to homotopy previously and cohomology rings of more general closed (and simply-connected) manifolds via construction of these maps. In the present paper, as a more precise work, we capture so-called (triple) Massey products in this way.
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.
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.
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.
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 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…
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…
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.
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…
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.
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.
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 …
Special generic maps are smooth maps at each singular point of which we can represent as $(x_1, \cdots, x_m) \mapsto (x_1,\cdots,x_{n-1},\sum_{k=n}^{m}{x_k}^2)$ for suitable coordinates. Morse functions with exactly two singular points on homotopy spheres and canonical projections of unit spheres are special generic. They are known to restrict the topologies and the differentiable structures of the manifolds in various situations. On the other hands, various manifolds admit such maps. This article first presents a special generic map on a $7$-dimensional manifold and the image. This results al…
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 …
