The stable mapping class group of simply connected 4-manifolds
Jeffrey Giansiracusa
备注：
22 pages, 3 figures; v2 - strengthened results on Dehn twists, added acknowledgements, corrected comments on relation to Hermitian K-theory; v3 - minor corrections and added section 8 on operads, final version to appear in Crelle's Journal; v4 - added reference
We consider mapping class groups Γ(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic K-theory of the integers. We also comment on versions of these results for simply connected spin 4-manifolds. Finally, we observe that a related 4-manifold operad detects infinite loop spaces.
We attach copies of the circle to points of a countable dense subset $D$ of a separable metric space $X$ and construct an earring space $E(X,D)$. We show that the fundamental group of $E(X,D)$ is isomorphic to a subgroup of the Hawaiian earring group, if the space $X$ is simply-connected and locally simply-connected. In addition if the space $X$ is locally path-connected, the space $X$ can be recovered from the fundamental group of $E(X,D)$.
We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with well-behaved skeleta and some Poincaré duality features.
We give necessary and sufficient conditions for a closed smooth 6-manifold N to be diffeomorphic to a product of a surface F and a simply connected 4-manifold M in terms of basic invariants like the fundamental group and cohomological data. Any isometry of the intersection form of M is realized by a self-diffeomorphism of M x F.
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.
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 \…
Diffeomorphism types of simply connected $3$-dimensional Mori fibre spaces
School of Mathematics, Shandong University（山东大学数学学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院；中国科学院大学）；University of Chinese Academy of Sciences（广西民族大学数学科学学院）；School of Mathematical Sciences, Guangxi Minzu University
We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.
We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with nontrivial Seiberg-Witten invariants.
We describe a method for counting the number of $1$-connected trivalent $2$-stratifolds with a given number of singular curves and $2$-manifold components.
In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.
This is primarily an exposition, combining work of several authors (Curtis, Hsiang, Freedman, Stong, Matveyev, and Bizaca), of the proof that a smooth 5-dimensional h-cobordism between simply connected 4-manifolds is a product off of a contractible piece which itself is diffeomorphic to the 5-ball.
We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.
The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. Using this correspondence and combinatorial arguments, we prove that a simply connected 6-dimensional torus manifold with vanishing odd degree cohomology is equivariantly diffeomorphic to the 6-dimensional sphere or an equivariant connected sum of copies of 6-dimensional quasitor…
We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension-2 spines in simply-connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.
We characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.
Several authors have recently attempted to show that the intersection of three simply connected subcontinua of the plane is simply connected provided it is non-empty and the intersection of each two of the continua is path connected. In this note we give a very short complete proof of this fact. We also confirm a related conjecture of Karimov and Repovs.
It is well known that Sullivan showed that the mapping class group of a simply connected high-dimensional manifold is commensurable with an arithmetic group, but the meaning of "commensurable" in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan's result: rather we provide a gloss for it.
For each rational homology 3-sphere $Y$ which bounds simply connected definite 4-manifolds of both signs, we construct an infinite family of irreducible rational homology 3-spheres which are homology cobordant to $Y$ but cannot bound any simply connected definite 4-manifold. As a corollary, for any coprime integers $p,q$, we obtain an infinite family of irreducible rational homology 3-spheres which are homology cobordant to the lens space $L(p,q)$ but cannot obtained by a knot surgery.
Trivalent $2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where three sheets meet. We obtain a classification of $1$-connected $2$-stratifolds in terms of their associated labeled graphs and develop operations that will construct from a single vertex all graphs that represent $1$-connected $2$-stratifolds.
We construct a new family of simply connected minimal complex surfaces with $p_g=1$, $q=0$, and $K^2=8$ using a $\mathbb{Q}$-Gorenstein smoothing theory.
We consider simply-connected $4$-manifolds admitting Lefschetz fibrations over the $2$-sphere. We explicitly construct nonhyperelliptic and hyperelliptic Lefschetz fibrations of genus $4$ on simply-connected $4$-manifolds which are exotic symplectic $4$-manifolds in the homeomorphism classes of $\mathbb{C}P^{2}\#8\overline{\mathbb{C} P^{2}}$ and $\mathbb{C} P^{2}\#9\overline{\mathbb{C} P^{2}}$, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to $18$ for $g=3$ when such fibrations are…
We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected.
We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group G. In each case, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth s-cobordism and hence are topologically equivalent for good groups.
Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano continuum for any n>0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting with the circle $\mathbb{S}^1$, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum.
A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.
In this paper we construct non-simply connected contact manifolds $M$ of dimension $\geq5$ such that $M\times S^1$ does not admit a symplectic structure.
The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions. We then show that for simply connected 5--manifolds these restrictions are necessary and sufficient.
For each pair $(e,σ)$ of integers satisfying $2e+3σ\ge 0$, $σ\leq -2$, and $e+σ\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $σ$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).
As the sequel to [5, 7], we construct a simply connected minimal complex surface of general type with p_g = 0 and K^2 = 4 by using a rational blow-down surgery and Q-Gorenstein smoothing theory.
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.
We introduce dual matroids of 2-dimensional simplicial complexes. Under certain necessary conditions, duals matroids are used to characterise embeddability in 3-space in a way analogous to Whitney's planarity criterion. We further use dual matroids to extend a 3-dimensional analogue of Kuratowski's theorem to the class of 2-dimensional simplicial complexes obtained from simply connected ones by identifying vertices or edges.
We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this addresses the Remark 4.79 of Besse.
In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{\mathbf CP}^2}$ admits an exotic smooth structure.
This paper is the last paper in a series of five papers. Building on earlier papers in this series, we prove an analogue of Kuratowski's characterisation of graph planarity for three dimensions. More precisely, a simply connected 2-dimensional simplicial complex embeds in 3-space if and only if it has no obstruction from an explicit list of obstructions. This list of obstructions is finite except for one infinite family.
We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The Fibonacci numbers make a cameo appearance.
We prove that any closed simply-connected smooth 4-manifold is 16-fold branched covered by a product of an orientable surface with the 2-torus, where the construction is natural with respect to spin structures. In particular this solves Problem 4.113(C) in Kirby's list. We also discuss analogous results for other families of 4-manifolds with infinite fundamental groups.
For 5 <= k <= 8 we show that the infinite family of exotic smooth structures on CP^2# k(-CP^2) can be achieved by 1/n - surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP^2# k(-CP^2).
We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod p technique and defor…
We show that for every sequence $(n_i)$, where each $n_i$ is either an integer greater than 1 or is $\infty$, there exists a simply connected open 3-manifold $M$ with a countable dense set of ends $\{e_i\}$ so that, for every $i$, the genus of end $e_i$ is equal to $n_i$. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in $S^3$. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus.
We prove the existence of $(20-2K^2)$-dimensional families of simply-connected surfaces with ample canonical class, $p_g=1$, and $1 \leq K^2 \leq 9$, and we study the relation with configurations of rational curves in K3 surfaces via $\mathbb Q$-Gorenstein smoothings. Our surfaces with $K^2=7$ and $K^2=9$ are the first surfaces known in the literature, together with the existence of a $4$-dimensional family for $K^2=8$.
