Alex N. Dranishnikov, Yuli B. Rudyak
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.
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For each composite number $n\ne 2^k$, there does not exist a single connected closed $(n+1)$-manifold such that any smooth, simply-connected, closed $n$-manifold can be topologically flat embedded into it. There is a single connected closed 5-manifold $W$ such that any simply-connected, 4-manifold $M$ can be topologically flat embedded into $W$ if $M$ is either closed and indefinite, or compact and with non-empty boundary.
We determine the number of distinct fibre homotopy types for the gauge groups of principal $Sp(2)$-bundles over a closed, simply-connected four-manifold.
In this paper, we show that for a simply connected CW complex $Y$ with $H^{*}(Y;\mathbb{Q})$ of finite dimension, if $H^{*}(Y;\mathbb{Q})$ is concentrated in degrees $\leq 3$, then the rationalization $Y_\mathbb{Q}$ is formal. As an application, we show that the spatial realizations of simply connected Sullivan algebras with rational homology concentrated in degrees $\leq 3$ are formal.
We construct examples of non-formal simply connected and compact oriented manifolds of any dimension bigger or equal to 7.
We introduce a rational homotopy invariant P of a topological space, which is a quintic tensor on the cohomology. For n > 1, formality of a closed (n-1)-connected manifold of dimension up to 5n-2 is equivalent to vanishing of P and the Bianchi-Massey tensor introduced by Crowley and the second author arXiv:1505.04184. We show also that elements of the group of closed simply-connected spin 8-manifolds with the cohomology of an r-fold connected sum of S^2 x S^6 are determined up to torsion by the value of P.
A non-formal simply connected compact symplectic manifold of dimension 8 is constructed.
Iterated finite group actions on closed connected aspherical manifolds
本文通过自由迭代作用和迭代离散对称度研究非退化流形的刚性，引入迭代作用长度概念并研究nil流形、sol流形和局部对称空间的性质。
Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal $G$-bundles over $M$ when $π_1(M)$ is: (1)~$\mathbb{Z}^{*m}$, (2)~$\mathbb{Z}/p^r\mathbb{Z}$, or (3)~$\mathbb{Z}^{*m}*(*^n_{j=1}\mathbb{Z}/p_j^{r_j}\mathbb{Z})$, where $p$ and the $p_j$'s are odd primes.
7-dimensional closed and simply-connected manifolds have been attractive as central and explicit objects in algebraic topology and differential topology of higher dimensional closed and simply-connected manifolds, which were studied actively especially in the 1950s--60s. Attractive studies of the class of these $7$-dimensional manifolds were started by the discovery of so-called exotic spheres by Milnor. It has influenced on the understanding of higher dimensional closed and simply-connected manifolds via algebraic and abstract objects. Recently this class is studied via more concrete notions …
Characterizing certain classes of $6$-dimensional closed and simply-connected manifolds via special generic maps
本文研究6维闭单连通流形到欧氏空间的特殊generic映射存在性，给出新充要条件，拓展了Reeb定理与Nishioka的5维结果，推进了高维流形拓扑分类研究。
We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Loeh up to dimension five. More precisely, we show that: (1) every simply connected, closed four-manifold admits a branched double covering by a product of the circle with a connected sum of copies of $S^2 \times S^1$, followed by a collapsing map; (2) every simply connected, closed five-manifold admits a branched double covering by a product of the circle with a connected sum of copies of $S^3 \…
Failure of Ordinary TQFTs to Distinguish Homotopy Type
本文构造反例表明，普通TQFT无法区分某些具有无限π2的单连通闭6维流形，并推广到所有高维及非单连通5维流形，证明普通TQFT在区分同伦型上存在根本局限。
Closed (and simply-connected) manifolds whose dimensions are greater than 4 are classified via sophisticated algebraic and abstract theory such as surgery theory and homotopy theory. It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. However, the latter work via geometric and constructive ways is not so difficult. The assumption that the dimensions are not high enables us to handle the manifolds via diagrams for example. It is difficult to study higher dimensional manifolds in these ways, although it is natural and important. In the present paper, we present such studi…
The class of special generic maps is a natural class of smooth maps containing Morse functions on spheres with exactly two singular points and canonical projections of unit spheres. We find new restrictions on such maps on $6$-dimensional or higher dimensional closed and simply-connected manifolds into ${\mathbb{R}}^5$. Spheres which are not diffeomorphic to unit spheres do not admit such maps whose codimensions are negative in considerable cases. They restrict the homeomorphism and the diffeomorphism types of the manifolds in general. On the other hands, some elementary manifolds admit specia…
An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allow us to prove that all, but one, of the known examples of simply-connected inflexible manifol…
Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of 2 points in $M$, $F(M,2)$. When $M$ is even dimensional, we prove that the rational homotopy type of $F(M,2)$ depends only on the rational homotopy type of $M$. When the dimension of $M$ is odd, for every $x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $C(x)$. We prove that for some $x \in H^{n-2} (M; \mathbb{Q})$, $C(x)$ encodes completely the rational homotopy type of $F(M,2)$. For some class of manifolds, we show that we can…
We study perfect discrete Morse functions on closed oriented n-dimensional manifolds. We show how to compose such functions on connected sums of closed oriented manifolds and how to decompose on connected sums of closed oriented surfaces.
This article presents families of 7-dimensional closed and simply-connected manifolds and fold maps on them such that squares of 2nd integral cohomology classes may not be divisible by 2. Fold maps are higher dimensional versions of Morse functions. The author has launched and been challenging the following new area: geometric and constructive studies of higher dimensional, closed and simply-connected manifolds. They are central objects in classical algebraic topology and differential topology. They were classified via algebraic and abstract objects in the last century and their understanding …
Building on work of Stolz, we prove for integers $0 \le d \le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal-Wi…
We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$ or $H^*(S^2\times S^5)$, this $7$-manifold admits a Riemannian metric with positive Ricci curvature.
In this paper, we continue our study of closed models defined in categories of graphs. We construct a closed model defined in the cat-egory of directed graphs which characterizes the strongly connected components. This last notion has many applications, and it plays an important role in the web search algorithm of Brin and Page, the foun-dation of the search engine Google. We also show that for this closed model, Euler graphs are particular examples of cofibrant objects. This enables us to interpret in this setting the classical result of Euler which states that a directed graph is Euleurian i…
Let $\{V_{i}\}_{i=1}^{n}$ be a finite family of closed subsets of a plane or a sphere $S^{2}$, each homeomorphic to the two-dimensional disk. In this paper we discuss the question how the boundary of connected components of a complement $\rr^{2} \setminus \bigcup_{i=1}^{n} V_{i}$ (accordingly, $S^{2} \setminus \bigcup_{i=1}^{n} V_{i}$) is arranged. It appears, if a set $\bigcup_{i=1}^{n} \Int V_{i}$ is connected, that the boundary $\partial W$ of every connected component $W$ of the set $\rr^{2} \setminus \bigcup_{i=1}^{n} V_{i}$ (accordingly, $S^{2} \setminus \bigcup_{i=1}^{n} V_{i}$) is home…
Formality of k-connected spaces in 4k+3 and 4k+4 dimensions
本文利用s-形式性扩展Miller定理的界，证明在特定Betti数条件下，具有硬Lefschetz性质的k-连通高维流形是形式的，并给出辛几何与特殊和乐流形中的例子。
Round fold maps are smooth maps on closed manifolds which are locally represented as the product maps of Morse functions and identity maps on open disks and whose singularity is realized as concentrically embedded spheres. The author previously introduced such maps. Our paper presents round fold maps on some $7$-dimensional simply-connected manifolds whose cohomology rings are isomorphic to that of the product of the $2$-dimensional complex projective space and a $3$-dimensional sphere. Such manifolds have been studied precisely by Wang and round fold maps on spin manifolds in these manifolds …
We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.
Morse functions with exactly two singular points on spheres and canonical projections of spheres belong to the class of a certain good class of smooth maps: special generic maps. We mainly investigate information on cohomology of closed and simply-connected manifolds admitting such maps into the $5$-dimensional Euclidean spaces by investigating the embedded curves and submanifolds and their preimages. Studies on homology groups for ones into the Euclidean spaces (whose dimensions are lower than $5$ in most cases) have been pioneered by Saeki and Sakuma since 1990s and later by Nishioka and Wra…
Let $ΛX$ be the free loop space on a simply connected finite $CW$-complex $X$ and $β_{i}(ΛX;\Bbbk)$ be the cardinality of a minimal generating set of $H^{i}(ΛX;\Bbbk)$ for $\Bbbk$ to be a commutative ring with unit. The sequence $ β_{i}(ΛX;\Bbbk) $ grows unbounded if and only if $\tilde {H}^{\ast}(X;\Bbbk)$ requires at least two algebra generators. This in particular answers to a long standing problem whether a simply connected closed smooth manifold has infinitely many geometrically distinct closed geodesics in any Riemannian metric.
The Yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature. Recently, Petean showed that the Yamabe invariant is non-negative for all closed simply connected manifolds of dimension $\ge 5$. We extend this to show that Yamabe invariant is non-negative for all closed manifolds of dimension $\ge 5$ with fundamental group of odd order having all Sylow subgroups abelian. The main new geometric input is a way of studying the Yamabe invariant on Toda brackets. A similar method of proof shows that all closed manifolds of dimension $\ge 5$…
The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $G\cong K\times \mathbb{R}^d$ with $K\le G$ maximal compact. We prove that the characteristic indices of closed connected subgroups are dominated by those of the ambient groups.
We prove that an injective map $f:X\to Y$ between connected metrizable spaces $X,Y$ is continuous if for every connected subset $C\subset X$ the image $f(C)$ is connected and one of the following conditions is satisfied: (1) $Y$ is a 1-manifold and $X$ is compact; (2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$; (3) $Y$ is a 3-manifold and $X$ is a simply-connected closed $n$-manifold of dimension $n\ge 3$. This gives a partial answer to a problem of Willie Wong, posed on Mathoverflow.
We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexisten…
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。
Let M be a closed (n-1)-connected 2n-dimensional smooth manifold with n > 2. In terms of the system of invariants for such manifolds introduced by Wall, we obtain necessary and sufficient conditions for M to admit an almost complex structure.
Universal covers of non-negatively curved manifolds and formality
本文证明闭光滑流形若容许非负Ricci曲率度量且万有覆盖形式，则流形本身形式，并去掉可定向性假设，推广到部分非闭流形。
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calculated. We show that for a generic metric on such a smooth four manifold with second Betti number at least three, the number of geometrically distinct periodic geodesics of length at most l grow exponentially as a function of l. The number of closed Reeb orbits of length at most l o…
We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rational cohomology $S^2\times {\mathbb C}P^2$. An infinite family of six-dimensional simply connected biquotients whose second Betti number is three, different from Totaro's biquotients, is considered and it is proved that none of biquotient from this family is geometrically formal…
Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the conve…
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