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We show that for any connected smooth manifold $M$ of dimension different from $3$ the restriction of the compact-open topology to the diffeomorphism group of $M$ is minimal, i.e. the group does not admit a strictly coarser Hausdorff group…

In this paper we give a full diffeomorphism characterization of compact simply connected cohomogeneity one manifolds in dimension six.

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or discompacit\'e in…

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the…

A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many…

Our goal is to identify curvature conditions that distinguish Euclidean space in the case of open, contractible manifolds and the disk in the case of compact, contractible manifolds with boundary. First, we show that an open manifold that…

In this paper, we classify simply connected closed smooth $13$-dimensional manifolds whose cohomology ring is isomorphic to that of $\mb{CP}^3\times S^7$, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a…

For a compact $(2n+1)$-dimensional smooth manifold, let $\mu_M : B Diff_\partial (D^{2n+1}) \to B Diff (M)$ be the map that is defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and…

An odd-dimensional differentiable manifold is called \emph{holomorphically fillable} if it is diffeomorphic to the boundary of a compact strongly pseudoconvex complex manifold, \emph{Stein fillable} if this last manifold may be chosen to be…

A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like…

We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the…

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary…

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic…

We construct smooth manifolds with order two $\pi_1$ and even intersection forms which are irreducible, meaning they do not decompose into non-trivial connected sums. Their intersection forms being even implies that their universal covers…

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic and Stein fillings can be classified up to…

In this note we give a direct method to classify all stable forms on $\R^n$ as well as to determine their automorphism groups. We show that in dimension 6,7,8 stable forms coincide with non-degnerate forms. We present necessary conditions…

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair…

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely…

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In this note, we first construct an $8$-dimensional compact smooth manifold whose unique…

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the…