Soren Galatius, Oscar Randal-Williams
We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises 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 these moduli spaces in a range of degrees.
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The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions.
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 $n \geq 3$.
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 $S^{n}\times S^{n}$.
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 $2n \geq 6$.
We describe partial semi-simplicial resolutions of moduli spaces of surfaces with tangential structure. This allows us to prove a homological stability theorem for these moduli spaces, which often improves the known stability ranges and give explicit stability ranges in many new cases. In each of these cases the stable homology can be identified using the methods of Galatius, Madsen, Tillmann and Weiss.
A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and isotopy class. As well as for unparametrised submanifolds, we prove this also for partially-parametrised submanifolds -- where a partial parametrisation may be thought of as a superposition of parametrisations related by a fixed subgroup of the mapping …
Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces. The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. As a consequence the map to $K$-theory defined by the action of t…
Following the argument for diffeomorphisms by Galatius and Randal-Williams, we prove that homeomorphisms of 1-connected manifolds of even dimension at least 6 exhibit homological stability. We deduce similar results for PL homeomorphisms and homeomorphisms as a discrete group.
The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts of open-closed string theory, we give an estimate of rational homological dimension of Riemann suraces with possible boundary and marked points(can lie on both interior and boundary). We hope it will have applications in open-closed theory, for example, open-closed Gromov-Witten theory in the future.
We show rational homological stability for the homotopy automorphisms and block diffeomorphims of iterated connected sums of products of spheres. The spheres can have different dimension, but need to satisfy a certain connectivity assumption. The main theorems of this paper extend homological stability results for automorphism spaces of connected sums of products of spheres of the same dimension by Berglund and Madsen.
We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both models.
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.
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 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。
By proving that several new complexes of embedded disks are highly connected, we obtain several new homological stability results. Our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed $E_n$-algebras. Using this, we prove a special case of a conjecture of Vakil and Wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally.
Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least dimension five and is simply-connected. We introduce a notion of tangential structure for subsurfaces and give a general criterion for when the space of subsurfaces with tangential structure satisfies homological stability provided that the manifold is simply-connected and has dimension $n\geq 5$. Examples of tangential structures such that the spaces of su…
We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be …
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension 2n…
Homological stability of diffeomorphism groups of high dimensional manifolds via $E_k$-algebras
本文利用 $E_k$-代数研究了高维流形微分同胚群的同调稳定性，改进了有理系数下的稳定性结果，并提出了量化同调稳定性这一新型稳定性结论。
We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.
We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditions are sharp when the background manifold is a two-dimensional sphere, the classical counterexample in …
Secondary Homological Stability for Unordered Configuration Spaces
本文证明了连通流形无序构型空间同调的次同调稳定性，通过构造新的链层级稳定化映射解决了闭流形情形下缺乏诱导稳定性映射及同调呈周期性的难题。
In [Chu12], Church used representation stability to prove that the space of configurations of distinct unordered points in a closed manifold exhibit rational homological stability. A second proof was also given by Randal-Williams in [RW11] using transfer maps. We give a third proof of this fact using localization and rational homotopy theory. This gives new insight into the role that the rationals play in homological stability. Our methods also yield new information about stability for torsion in the homology of configuration spaces of points in a closed manifold.
We define bounded generation for $E_n$-algebras in chain complexes and prove that for $n \geq 2$ this property is equivalent to homological stability. Using this we prove a local-to-global principle for homological stability, which says that if an $E_n$-algebra $A$ has homological stability (or equivalently the topological chiral homology $\int_{R^n} A$ has homology stability), then so has the topological chiral homology $\int_M A$ of any connected non-compact manifold $M$. Using scanning, we reformulate the local-to-global homological stability principle in a way that also applies to compact …
Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n \ge 3$.
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.
We study the space of oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a "scanning map" which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees. Our results are analogous to McDuff's theorem on configuration spaces, extended from 0-manifolds to 2-manifolds.
In Appendix A of his article on rational functions, Segal proved homological stability for configuration spaces with a stability slope of 1/2. This was later improved to a slope of 1 by Church and Randal-Williams if one works with rational coefficients and manifolds of dimension at least $3$. In this note we prove that the stability slope of 1 holds even with Z[1/2] coefficients, and clarify some aspects of Segal's proof for topological manifolds.
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered kn…
Homological monotonicity for configuration spaces of manifolds
该研究针对流形无序配置空间的同调性质，证明其每个次数下的同调维数单调递增，且该单调性不依赖流形的微分结构与可定向性，补充了经典稳定同调结论之外的不稳定域性质。
Bredon homological stability for configuration spaces of $G$-manifolds
本文研究了G-流形的无序配置空间的Bredon同调稳定性，定义了equivariant稳定映射，并在Dedekind群条件下证明了同调稳定性结果。
We prove a homological stability theorem for certain complements of symmetric spaces. This is a variant of a conjecture by Vakil and Matchett Wood for subspaces of $\mathrm{Sym}^n(X)$ where $X$ is an open manifold admitting a boundary. To do this we prove a homological stability result for a type of "coloured" configuration space by adding points of the same colour.
Homology stability for asymptotic monopole moduli spaces
该研究证明了带框狄拉克磁单极和理想磁单极两类渐近模空间的同调稳定性，通过配置空间的一般同调稳定性结果实现，并揭示了它们作为经典模空间部分紧化边界超曲面的结构。
Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory
本文发展了主 G-丛的前推运算理论，在 G=BU(1) 及扭曲 K-理论中构造并分类了由投影 Euler 和秩运算生成的稳定前推运算，并在交换 H-空间的同调上构造了应用于枚举几何穿墙公式的分次李代数结构。
We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Forman's discrete Morse theory for CW-complexes and generalizes Forman and Chari's results on the face posets of regular CW-complexes. We also introduce a homological variant of the theory that can be used to study the topology of triangulable homology manifolds by means of their order triangulations.
This paper consists of two related parts. In the first part we give a self-contained proof of homological stability for the spaces C_n(M;X) of configurations of n unordered points in a connected open manifold M with labels in a path-connected space X, with the best possible integral stability range of 2* \leq n. Along the way we give a new proof of the high connectivity of the complex of injective words. If the manifold has dimension at least three, we show that in rational homology the stability range may be improved to * \leq n. In the second part we study to what extent the homology of the …
$C$-triviality of manifolds of low dimensions
本文研究了低维闭光滑流形的C-平凡性，通过谱序列和微分的阶数进行同调分类，7维时给出可定向和不可定向流形的条件。
We survey recent work on moduli spaces of manifolds with an emphasis on the role played by (stable and unstable) homotopy theory. The theory is illustrated with several worked examples.
We discuss various aspects of `braid spaces' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomological dimension of braid spaces. This is related to some sharp and useful connectivity bounds that we establish for the reduced symmetric products of any simplicial complex. Our methods are geometric and exploit a dual version of configuration spaces given in terms of truncated …
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…
In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant $\e$, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H…
We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let l>2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of F_q(t) have the l-part of their class group isomorphic to A.
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.
We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations. In the first part, we introduce and study plant complexes, a large new class of simplicial complexes, generalizing the arc complex on a surface with marked points. In the second part, we generalize a result by Ellenberg-Venkatesh-Westerland by showing that homological stabilization…
We prove Conjecture F from [VW12] which states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. Moreover, we generalize this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.
Motivic homological stability of configuration spaces
本文证明了流形中点的构型空间的经典同调稳定性结果可被提升至motivic上同调。
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…
We have recently proved a homological stability theorem for moduli spaces of r-Spin Riemann surfaces, which in particular implies a Madsen--Weiss theorem for these moduli spaces. This allows us to effectively study their stable cohomology, and to compute their stable rational cohomology and their integral Picard groups. Using these methods we give a complete description of their integral Picard groups for genus at least 9 in terms of geometrically defined generators, and determine the relations between them.
We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal $d$-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space $S(Y)$, with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories \HL and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev.
Conjecture F from [VW12] states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.
Homological smoothness and Deligne resolution for tropical fans
本文证明热带扇的同调光滑性具有 T-稳定性，从而推出拟线性扇及其局部建模的热带簇为热带同调流形，并建立 Deligne 权谱序列的热带类比。
Morse-Bott cohomology from homological perturbation theory
该研究在最小横截性假设下基于同调扰动理论构建Morse-Bott理论的上链复形，讨论不同构造的关联，完成等变情形构造并验证性质，在闭流形情形验证其恢复正则上同调，还提出与polyfold理论结合的计划。
We consider moduli spaces of $d$-dimensional manifolds with embedded particles and discs. In this moduli space, the location of the particles and discs is constrained by the $d$-dimensional manifold. We will compare this moduli space with the moduli space of $d$-dimensional manifolds in which the location of such decorations is no longer constrained, i.e. the decorations are decoupled. We generalise work by Bödigheimer--Tillmann for oriented surfaces and obtain new results for surfaces with different tangential structures as well as to higher dimensional manifolds. We also provide a generalisa…
We construct a zig-zag from the once delooped space of pseudoisotopies of a closed $2n$-disc to the once looped algebraic $K$-theory space of the integers and show that the maps involved are $p$-locally $(2n-4)$-connected for $n>3$ and large primes $p$. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik--Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa's stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification …
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.
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…
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…
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.
Twisted homological stability for handlebody mapping class groups
该研究证明了手柄映射类群的扭转同调稳定性，并通过引入系数双系统，扩展了手柄稳定性结果，应用于三维手柄模空间的同调稳定性。
Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homolog…
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