Craig R. Guilbault
In this note we present a characterization of those open n-manifolds (n>4), whose products with the real line are homeomorphic to interiors of compact (n+1)-manifolds with boundary.
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In this note we prove that equivariantly homeomorphic quasitoric manifolds are diffeomorphic. As a consequence we show that up to finite ambiguity the diffeomorphism type of certain quasitoric manifolds $M$ is determined by their cohomology rings and first Pontrjagin classes.
In this paper we construct a minimal symplectic 4-manifold and prove it is homeomorphic but not diffeomorphic to CP^2 # 3(-CP^2)
For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. These manifolds are the fi…
An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to {\bf R}$^3$ or the group is a free product of infinite cyclic groups and infinite closed surface groups. Given any such finitely generated group uncountably many $M$ are constructed with that fundamental group such that their universal covers are {\bf R}$^2$-irreducible, are not…
We present some classification results for quasitoric manifolds (M) with (p_1(M)=-\sum a_i^2) for some (a_i\in H^2(M)) which admit an action of a compact connected Lie-group (G) such that (\dim M/G \leq 1). In contrast to Kuroki's work we do not require that the action of (G) extends the torus action on (M).
Stably exotic fillings of 3-manifolds
该研究探讨三维流形界定四维流形的稳定奇异填充问题，证明闭定向三维流形及部分非定向三维流形存在此类填充，而含双侧$\text{RP}^2$的三维流形不存在。
On manifolds homeomorphic to spheres
本文在Morse-Bott函数背景下证明了类似Reeb定理的结论，指出具有两个不同维数临界子流形的闭光滑流形必同胚于标准球面，并将此结果推广至实数、复数和四元数上的射影空间。
We are interested in contractible n-manifolds M which "split" as M = A union B where A,B, and A intersect B are all homeomorphic to Euclidean n-space (such M are called open n-splitters) or A,B, and A intersect B are all homeomorphic to the n-dimensional unit ball (such M are called closed n-splitters). We introduce a closed 4-splitter M from which we construct an infinite collection of distinct closed 4-splitters and an uncountable set of open 4-splitters.
It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the authors \cite{BCS}. The main result of this paper proves the existence of local sections in a Serre fibration with all fibers homeomorphic to some fixed compact three-dimensional manifold.
This paper describes how to recover the topology of a closed manifold $M$ from a good Morse function $f$ on $M$. The essential method was suggested by Cohen, Jones and Segal. They constructed a topological category $C_{f}$ and claimed that the classifying space $BC_{f}$ is homeomorphic to $M$. We prove it from a different viewpoint with them using a cell decomposition of $M$ associated to $f$. The cell complex $M_{f}$ equipped with the decomposition induces a topological category $C(M_{f})$ whose classifying space $BC(M_{f})$ is homeomorphic to $M$. We show that $C(M_{f})$ is isomorphic to $C_…
We show that any open aspherical manifold of dimension n>3 is tangentially homotopy equivalent to an n-manifold whose universal cover is not homeomorphic to the Euclidean space.
We show that there is no algorithm to decide whether or not a given 4-manifold is homeomorphic to the connected sum of 12 copies of S^2 \times S^2.
A classical result by Pachner states that two $d$-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly $(d + 1)$-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds wi…
Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that Lambda(M)=D(M)(2-D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Lambda(M)=1 and D(M)=1 in terms of the characteristic submanifold of the incompressib…
It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. Results of this paper extend Whitney theorem to the case when all fibers are homeomorphic to a given compact two-dimensional manifold.
In this paper we investigate the relationship between isotopy classes of knots and links in S^3 and the diffeomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of diffeomorphism types of manifolds homeomorphic to the K3 surface.
In this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to $M \times [0, \infty)$ where $M$ is a closed Euclidean manifold. These are classified in [2]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of $π_1(M)$. It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on $…
Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer $n\geq 3$. We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic smooth structures when $M$ is $\CP#4\CPb$ or $3\CP# k \CPb$ for $k=6,8,10$.
We show examples of pairs of smooth, compact, homeomorphic 4-manifolds, whose diffeomorphism types are distinguished by the topology of the singular sets of smooth stable maps defined on them. In this distinction we rely on results from Seiberg-Witten theory.
Suppose $X$ and $Y$ are compact connected topological 4-manifolds with fundamental group $π$. For any $r \geqslant 0$, $Y$ is $r$-stably homeomorphic to $X$ if $Y \# r(S^2 \times S^2)$ is homeomorphic to $X \# r(S^2\times S^2)$. How close is stable homeomorphism to homeomorphism? When the common fundamental group $π$ is virtually abelian, we show that large $r$ can be diminished to $n+2$, where $π$ has a finite-index subgroup that is free-abelian of rank $n$. In particular, if $π$ is finite then $n=0$, hence $X$ and $Y$ are $2$-stably homeomorphic, which is one $S^2 \times S^2$ summand in exce…
Gabai showed that the Whitehead manifold is the union of two submanifolds each of which is homeomorphic to $\mathbb R^3$ and whose intersection is again homeomorphic to $\mathbb R^3$. Using a family of generalizations of the Whitehead Link, we show that there are uncountably many contractible 3-manifolds with this double 3-space property. Using a separate family of generalizations of the Whitehead Link and using an extension of interlacing theory, we also show that there are uncountably many contractible 3-manifolds that fail to have this property.
We show Péter Csorba's conjecture that the graph homomorphism complex Hom(C_5,K_{n+2}) is homeomorphic to a Stiefel manifold, the space of unit tangent vectors to the n-dimensional sphere. For this a general tool is developed that allows to replace the complexes Hom(G, K_n) by smaller complexes that are homeomorphic to them whenever G is a graph for which those complexes are manifolds. The equivariant version of Csorba's conjecture is proved up to homotopy. We also study certain subdivisions of simplicial manifolds that are related to the interval poset of their face posets and their connectio…
Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M is homeomorphic to a sphere. (3)…
Suppose M is a connected, open, orientable, irreducible 3-manifold which is not homeomorphic to R^3. Given a compact 3-manifold J in M which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood V$ called an end reduction of M at J. It has some useful properties which allow one to extend to M various results known to hold for the more restrictive class of eventually end irreducible open 3-manifolds. In this paper we explore the relationship of V and M with regard to their fundamental groups and their covering spaces. In particular we give conditions under …
This article presents the constructions of new infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic and, from their construction, seem to be quite different, but cannot be distinguished by Seiberg-Witten invariants. Whether these manifolds are, or are not, actually diffeomorphic seems to be a very difficult question to answer. The most interesting of these constructions is a surgery that from certain symplectic 4-manifolds will produce nonsymplectic 4-manifolds. This is detected by calculations of Seiberg-Witten invariants. The su…
We exhibit the first examples of closed 7-dimensional Riemannian manifolds with holonomy G_2 that are homeomorphic but not diffeomorphic. These are also the first examples of closed Ricci-flat manifolds that are homeomorphic but not diffeomorphic. The examples are generated by applying the twisted connected sum construction to Fano 3-folds of Picard rank 1 and 2. The smooth structures are distinguished by the generalised Eells-Kuiper invariant introduced by the authors in arXiv:1406.2226.
We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-oriented taut foliation, are equivalent for graph manifold Z-homology 3-spheres.
We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal volume, or as the unique contractible $3$-manifold supporting a complete finite-volume Riemannian metric with Ricci curvature uniformly bounded from below. On the contrary, we show that in every dimension $n\geq 4$ there exists a contractible $n$-manifold with vanishing simplicial volume not homeomorphic to $\mathbb{R}^n$. We also compute the spectrum of the simpl…
Nonperiodic leaves of codimension one foliations
本文构造5维流形，说明其可作为C¹余维一或C∞余维二叶状结构的真叶，却不能成为紧致6维流形上C²余维一叶状结构的叶，并给出叶两两不同胚的极小不变集例子。
Contractible 3-manifold and Positive scalar curvature (II)
本文研究了正数量曲率下完备可缩3-流形同胚于$\mathbb{R}^{3}$的条件，通过探讨无穷远基群$π_{1}^{\infty}$与正数量曲率度量的关系，证明了当$π_{1}^{\infty}$平凡时该流形必同胚于$\mathbb{R}^{3}$。
We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.
In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_σ$-set $Σ_0\subset M$ which is universal meager in the sense that for each meager subset $B\subset M$ there is a homeomorphism $h$ of $M$ such that $h(B)\subset Σ_0$. Also we prove that any two universal meager $F_σ$-sets in $M$ are ambiently homeomorphic.
A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (quasi)toric manifold (in the sense of Davis-Januszkiewicz) over an n-cube with a semifree circle action and isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This …
This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For each countable free group a specific uncountable family of irreducible open 3-manifolds is constructed whose fundamental groups are isomorphic to the given group and whose universal covering spaces are in this class a…
Heegaard Floer Surgery Formula and Cosmetic Surgeries
本文利用Heegaard Floer同调公式，证明了特定结的美容手术系数必须大于1，并且非整数手术不会与原流形同胚。
The main theorem of the paper shows that a smooth manifold which is homeomorphic to S^2xS^2 and has nonvanishing Ozsvath-Szabo invariant does not admit a perfect Morse function. I am withdrawing the paper because it is unclear to me if such a manifold exists.
Suppose M is a non-compact connected n-manifold without boundary, DD(M) is the group of C^\infty-diffeomorphisms of M endowed with the Whitney C^\infty-topology and DD_0(M) is the identity connected component of DD(M), which is an open subgroup in the group DD_c(M) \subset DD(M) of compactly supported diffeomorphisms of M. It is shown that DD_0(M) is homeomorphic to N \times IR^\infty for an l_2-manifold N whose topological type is uniquely determined by the homotopy type of DD_0(M). For instance, DD_0(M) is homeomorphic to l_2 \times IR^\infty if n = 1, 2 or n = 3 and M is orientable and irre…
Let M be an oriented compact positively curved 4-manifold. Let G be a finite subgroup of the isometry group of $M$. Among others, we prove that there is a universal constant C (cf. Corollary 4.3 for the approximate value of C), such that if the order of G is odd and at least C, then G is either abelian of rank at most 2, or non-abelian and isomorphic to a subgroup of PU(3) with a presentation \{A, B| A^m=B^n=1, BAB^{-1}=A^r, (n(r-1), m)=1, r\ne r^3=1(\text{mod}m) \}. Moreover, M is homeomorphic to CP^2 if G is non-abelian, and homeomorphic to S^4 or CP^2 if G is abelian of rank 2.
We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b^+ at most 3, with one possible exception when b^+=3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smal…
Smooth Structures on the product of 3-connected 8-manifolds with spheres
本文计算了3-连通8维流形与球面乘积的协合惯性群，并分类了同胚于该乘积的光滑流形，特别给出了$H^4(M;\mathbb{Z})=\mathbb{Z}$时与$M\times\mathbb{S}^1$同胚流形的微分同胚分类。
Classification of closed conformally flat Lorentzian manifolds with unipotent holonomy
本研究分类了n≥3维、PO(2,n)中单幂和乐的闭共形平坦洛伦兹流形，将其分为四类几何类型，明确了有限覆盖下的同胚形式，还进一步分类了其中容许本质共形流的两类流形。
Classes of branched surfaces extend the classes of surfaces or 2-dimensional manifolds satisfying suitable properties and defined in various manners. Reeb spaces of smooth maps of suitable classes into surfaces whose codimensions are negative are regarded as branched surfaces. They are the spaces of all connected components of preimages and natural quotient spaces of the manifolds of the domains. They are defined for general smooth maps and important topological objects in differential topology. They also play important roles in applied or applications of mathematics such as projections in dat…
In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd dimensional manifolds without boundary.
We present new open manifolds that are not homeomorphic to leaves of any C^0 codimension one foliation of a compact manifold. Among them are simply connected manifolds of dimension 5 or greater that are non-periodic in homotopy or homology, namely in their 2-dimensional homotopy or homology groups.
Let $X$ be a Hausdorff topological group and $G$ a locally compact subgroup of $X$. We show that $X$ admits a locally finite $σ$-discrete $G$-functionally open cover each member of which is $G$-homeomorphic to a twisted product $G\times_H S_i$, where $H$ is a compact large subgroup of $G$ (i.e., the quotient $G/H$ is a manifold). If, in addition, the space of connected components of $G$ is compact and $X$ is normal, then $X$ itself is $G$-homeomorphic to a twisted product $G\times_KS$, where $K$ is a maximal compact subgroup of $G$. This implies that $X$ is $K$-homeomorphic to the product $G/K…
A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to $S_g \times [0; 1]$ such that the boundary component $S_g \times \{ 0\}$ is geodesic. We prove that a compact Fuchsian manifold with convex boundary is uniquely determined by the induced path metric on $S_g \times \{1\}$. We do not put further restrictions on the boundary except convexity.
In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty \tilde M_2^n$, and (2) $\tilde M_1^n$ is homeomorphic to $\tilde M_2^n$. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension $\geq 6$ are topologically rigid. We include some straightforward consequences of topological rigidity (diagra…
We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the …
We construct smooth 4-manifolds that are homeomorphic but not diffeomorphic to the "cusp" and the "fishtail", which are certain thickened singular 2-spheres.
We show that for any $n\geq6$, for any $m$, we can find $m$ non-homeomorphic $n$-manifolds that can be mapped by cell-like maps onto the same space $X$.
Exotic aspherical 4-manifolds
本研究构造了同胚但不微分同胚的闭非球面光滑4-流形，为四维Borel猜想的光滑类比提供反例，核心方法是将反射群技巧应用于带边怪异4-流形对。
We show that the manifold *CP^2 # *RP^4, which is homotopy equivalent but not homeomorphic to CP^2 # RP^4, is in fact smoothable.
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