• Relatively hyperbolic spaces
• CAT
• PL manifolds

In this dissertation we will talk about manifolds of curvature bounded by above in the sense of $\CAT$, an approach originally formalized by Aleksandrov in \cite{Aleks}. The concept is general, but we are interested in seeing how much freedom it leaves in the manifold setting. Let us introduce some terminology to set the problem.

A geodesic is an isometric embedding of an interval into a metric space. We consider geodesic metric spaces, where for any pair of points there is a geodesic having them as extremes. Then, given points in a geodesic space, we can construct a geodesic triangle having them as vertices, and define a concept of curvature bounded by above by $\kappa$ by comparing triangles in the space with triangles having sides of the same length in $2$-dimensional model spaces of curvature $\kappa$; that is, the Euclidean space for $\kappa=0$, a suitably rescaled sphere for $\kappa>0$ and a suitably rescaled hyperbolic space for $\kappa<0$. A space is locally $\CAT \tonde{\kappa}$ if every point has a neighbourhood which is $\CAT \tonde{\kappa}$.

The (locally) $\CAT \tonde{\kappa}$ condition allows us to describe the curvature of a large class of metric spaces, of which the polyhedral metric complexes are notable representatives: these are spaces made up by polyhedra attached via isometries. If we restrict our attention to Riemannian manifolds, they are locally $\CAT \tonde{\kappa}$ if and only all of their sectional curvatures are less or equal than $\kappa$. We may ask ourselves, however, if for manifolds the requirement on a distance function compatible with the topology to be induced by a Riemannian metric is restrictive when talking about curvature. The answer is yes; and we will in fact exhibit a smooth closed $4$-manifold that supports a locally $\CAT \tonde{0}$ distance function but is not homeomorphic to any Riemannian manifold of non positive curvature.

We will follow the approach presented in \cite{DJLF}. The manifold will be the geometric realization of a cubical complex, i.e. polyhedral complexes made up by euclidean cubes. Cubical complexes are a very important tool in the Geometry of non positively curved spaces; an example is provided by the recent proof of Virtual Haken Conjecture. Their power comes from the fact that the locally $\CAT \tonde{0}$ condition on them can be verified in purely combinatorial terms. In particular, we have to look at a neighbourhood of a vertex, which is a cone on a simplicial complex called \emph{link} at that vertex.

We begin the exposition with a review of definitions on curvature of metric spaces. Then we construct, for a given complete $\CAT \tonde{0}$ space $X$ its boundary at infinity $\partial X$. We continue by describing a natural structure of metric space on finitely generated groups, and an equivalence relation called \emph{quasi isometry} useful in this context. Quasi isometry invariants become more evident when seen through a construction allowing us to see the space ``from the infinity'', namely the asymptotic cones. With the strength of this construction, we define hyperbolic groups, hyperbolic spaces, and generalizations of these concepts: relatively hyperbolic spaces and groups, and spaces with isolated flats. Hyperbolic spaces and spaces with isolated flats share a nice behaviour of their boundary at infinity when talking about quasi isometries which are \emph{quasi equivariant}, i.e. equivariant with respect to a cocompact properly discontinuous action of a group $G$; in particular, such quasi-isometry induces a homeomorphism between the boundaries at infinity.

We then pass to examine polyhedral complexes and prove the facts we need in the sequel, including the description of the curvature in combinatorial terms we already cited.

The cubical complex of the thesis of the main Theorem will be constructed starting from a link at any of its vertices. To be a $4$-manifold, this link has to be a triangulation of $\sfera^3$ with certain properties. We will define this properties and construct the required triangulation.

Finally, we pass to describing the manifold. There is a standard way, introduced by Davis in \cite{DavisBook}, to create a cubical complex once known the link we want its vertices to have. This construction also allows to give a concise description of the fundamental group of the manifold, which we will call $P$. If it was homeomorphic to a smooth Riemannian manifold $M$ of non positive curvature, the structure of the triangulation and thus of the fundamental group of both manifolds would allow us to conclude that the respective universal covers $\wtde{P}$ and $\wtde{M}$ would have both an isometrically embedded euclidean plane, with the embedding totally geodesic in the smooth case, and that the boundaries at infinity, topologically $\sfera^3$ in both cases, would be homeomorphic, with the homeomorphism taking the boundary of the flat found in $\wtde{P}$ to the boundary of the flat found in $\wtde{M}$. This will turn impossible: the boundary of the flat is a knot in $\sfera^3$, but it will be non trivial in the former case and trivial in the second.

In the final part, we will examine some difficulties we encounter when trying to generalize this result to the locally $\CAT \tonde{-1}$ setting.