On $Q$-manifolds bundles

Homological properties of graph manifolds

We consider the following properties of compact oriented irreducible graph-manifolds: to contain a $π_1$-injective surface (immersed, virtually embedded or embedded), be (virtually) fibered over $S^1$, and to carry a metric of nonpositive sectional curvature. It turns out that all these properties can be described from a unified point of view.

Homological stability for moduli spaces of high dimensional manifolds. II

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem for any simply-connected manifold of dimension $2n \geq 6$.

On generalized 3-manifolds which are not homologically locally connected

We show that the classical example $X$ of a 3-dimensional generalized manifold constructed by van Kampen is another example of not homologically locally connected (i.e. not HLC) space. This space $X$ is not locally homeomorphic to any of the compact metrizable 3-dimensional manifolds constructed in our earlier paper which are not HLC spaces either.

Homological Stability For Moduli Spaces of Odd Dimensional Manifolds

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to $\#^{g}(S^{n+1}\times S^{n})$, provided $n \geq 4$. This is an odd dimensional analogue of a recent homological stability result of S. Galatius and O. Randal Williams for the moduli space of manifolds diffeomorphic to $\#^{g}(S^{n}\times S^{n})$ for $n \geq 3$.

Virtual Homological Torsion of Closed Hyperbolic 3-manifolds

In this paper, we will use Kahn-Markovic's almost totally geodesic surfaces to construct certain $π_1$-injective 2-complexes in closed hyperbolic 3-manifolds. Such 2-complexes are locally almost totally geodesic except along a 1-dimensional subcomplex. Using Agol and Wise's result that fundamental groups of hyperbolic 3-manifolds are LERF and quasi-convex subgroups are virtual retract, we will show that closed hyperbolic 3-manifolds virtually contain any prescribed homological torsion: For any finite abelian group $A$, and any closed hyperbolic 3-manifold $M$, there exists a finite cover $N$ o…

Counting non-commensurable hyperbolic manifolds and a bound on homological torsion

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp. We also determine how the number of non-commensurable closed hyperbolic manifolds of dimension at least 3 and bounded diameter grows. The lower bound implies that the fraction of arithmetic manifolds tends to zero as the diameter goes up.

Prescribed virtual homological torsion of 3-manifolds

We prove that given any finite abelian group $A$ and any irreducible $3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold there exists a finite cover $M' \to M$ so that $A$ is a direct factor in $H_1(M',\mathbb{Z})$. This generalizes results of Sun and of Friedl-Herrmann.

Homological norms on nonpositively curved manifolds

We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.

光滑4-流形模空间的同调不稳定性

Homological instability for moduli spaces of smooth 4-manifolds

该研究证明单连通闭光滑4-流形模空间不满足同调稳定性，检测了不同4-流形模空间的同调差异，并借助Seiberg-Witten方程构造新的不稳定示性类以证明结论。

A note on rational homological stability for automorphisms of manifolds

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp(S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

Homological stability for moduli spaces of high dimensional manifolds. I

We prove a homological stability theorem for moduli spaces of simply-connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer's stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of degrees.

Homological stability for configuration spaces of manifolds

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B_n(M) satisfies classical homological stability: for each i, H_i(B_n(M); Q) is isomorphic to H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural exa…

Morse Homotopy and Homological Conformal Field Theory

By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This complements previous constructions due to R. Cohen et al., K. Costello and M. Kontsevich and is also the Morse theoretic counterpart to a conjectural construction of operations on the Lagrangian Floer homology of the zero section of a cotangent bundle, obtained by studying uncomp…

A homological estimate for the Thurston norm

We establish a new homological lower bound for the Thurston norm on 1-cohomology of 3-manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1-cohomology of 2-dimensional CW-complexes.

Homological Stability For The Moduli Spaces of Products of Spheres

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.

Analytic, Reidemeister and homological torsion for congruence three--manifolds

Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplicity results for the irreducible components of the space of cuspidal forms.

BPS Invariants for a Knot in Seifert Manifolds

We calculate homological blocks for a knot in Seifert manifolds when the gauge group is $SU(N)$. We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.

Homological invariance for asymptotic invariants and systolic inequalities

We show that the systolic constant, the minimal entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value on the real projective space) and derive an inequality between the minimal entropy and the systolic constant.

Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ…

Examples of homological torsion and volume growth

We provide examples of towers of covers of cusped hyperbolic 3-manifolds whose exponential homological torsion growth is explicitly computed in terms of volume growth. These examples arise from abelian covers of alternating links in the thickened torus. A corollary is that the spanning tree entropy for each regular planar lattice is given by the volume of a hyperbolic polyhedron.

Homological stability of topological moduli spaces

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilize as well. This generalizes a framework of Randal-Williams and Wahl for families of discrete…

Homological obstructions to string orientations

We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in term of the annihilator of the Thom class of the negative tangent bundle, and in terms of the vanishing of top-degree cohomology operations. We also indicate how the existence of such an operation-preserving duality implies the int…

流形 $\#_n \mathbb{CP}^2$ 的同调 Nielsen 实现问题

Homological Nielsen realization for the manifolds $\#_n \mathbb{CP}^2$

研究光滑正定4-流形 $M_n=\#_n\mathbb{CP}^2$ 上，有限等距群 $G\leq O(H_2(M_n;\mathbb{Z}))$ 何时可同构提升为有限定向保向微分同胚群，发现随着 $n$ 增大，随机子群和随机奇阶元素几乎必然不可实现。

4-流形上的有限群作用与等变丛

Finite group actions on 4-manifolds and equivariant bundles

本研究针对具有同调平凡局部线性循环群作用的4-流形，通过扭曲符号公式推导得出等变丛存在的充要条件，该条件体现为不动点数据与迷向表示间的同余关系。

A criterion for homeomorphism between closed Haken manifolds

In this paper we consider two connected closed Haken manifolds denoted by M^3 and N^3, with the same Gromov simplicial volume. We give a simple homological criterion to decide when a given map f: M^3-->N^3 between M^3 and N^3 can be changed by a homotopy to a homeomorphism. We then give a convenient process for constructing maps between M^3 and N^3 satisfying the homological hypothesis of the map f.

Fintushel--Stern knot surgery in torus bundles

Suppose that $X$ is a torus bundle over a closed surface with homologically essential fibers. Let $X_K$ be the manifold obtained by Fintushel--Stern knot surgery on a fiber using a knot $K\subset S^3$. We prove that $X_K$ has a symplectic structure if and only if $K$ is a fibered knot. The proof uses Seiberg--Witten theory and a result of Friedl--Vidussi on twisted Alexander polynomials.

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum build from the canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i--this is the non-oriented analogue of the Mumford conj…

On the moduli space of flat symplectic surface bundles

In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.

Homological stability of diffeomorphism groups

In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range.

Coincidences of homological densities, predicted by arithmetic

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences $\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ of spaces of $0$-cycles on manifolds $X$. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. The obstacle to proving such a theorem with current technology is how to deal…

Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4n+1

Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold $M$, on the linking form associated to the homology groups of $M$. In particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theo…

Witten-Reshetikhin-Turaev 不变量与拼合同调球的同调块

Witten-Reshetikhin-Turaev invariants and homological blocks for plumbed homology spheres

本文证明一类拼合三维流形上的 Gukov-Pei-Putrov-Vafa 猜想，通过比较有理函数与 false theta 函数渐近展开的新技术，由全纯性得到加权 Gauss 和消失。