Homotopy Types Of Manifolds Research Articles

Homotopy type of manifolds with partially horoconvex boundary

Let [Formula: see text] be an [Formula: see text]-dimensional compact connected manifold with boundary, [Formula: see text] a constant and [Formula: see text] an integer. We prove that [Formula: see text] supports a Riemannian metric with the interior [Formula: see text]-curvature [Formula: see text] and the boundary [Formula: see text]-curvature [Formula: see text], if and only if [Formula: see text] has the homotopy type of a CW complex with a finite number of cells with dimension [Formula: see text]. Moreover, any Riemannian manifold [Formula: see text] with sectional curvature [Formula: see text] and boundary principal curvature [Formula: see text] is diffeomorphic to the standard closed [Formula: see text]-ball.

Classification of Rational Homotopy Type for 8-Cohomological Dimension Elliptic Spaces

The different methods used to classify rational homotopy types of manifolds are in general fascinating and various (see [1,7,8]). In this paper we are interested to a particular case, that of simply connected elliptic spaces, denoted X, by discussing its cohomological dimension. Here we will the discuss the case when dimH*( Χ ;Q)=8 and χ(Χ)=0.

Bounded geometries and topology fluctuations in lattice quantum gravity

The authors exploit the coarse classification of Riemannian geometries provided by Gromov's pre-compactness theorem (1981) in order to give a rigorous characterization of the partition function of n-dimensional, (n>or=2), lattice quantum gravity. They prove that the resulting theory admits a continuum limit describing phase transitions between different homotopy types of manifolds, with phases parametrized by the fundamental group and by the (Whitehead) torsions of the manifolds sampled. They also show that the results obtained coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.

Mixing homotopy types of manifolds

Mixing homotopy types of manifolds

A Note on Tangent Bundles

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

On Pontrjagin classes and homotopy types of manifolds.

On Pontrjagin classes and homotopy types of manifolds.

On the Homotopy Type of Manifolds

1. The object of this note is to call attention to certain theorems, which follow very easily from some results due to E. Stiefel,' H. Seifert,2 Hassler Whitney,3 and myself.4 They refer to a class of manifolds which we call the class H, and are intended to throw light on the question, raised by W. Hurewicz,5 whether two closed manifolds of the same homotopy type are necessarily homeomorphic. The theorems depend both on M. H. A. Newman's6 theory of combinatorial equivalence, as re-developed by J. W. Alexander7 and carried further in S. S., and on theorems concerning differentiable manifolds. Therefore it is necessary to give a precise meaning to the term 'manifold'. By an n-dimensional manifold, M , we shall mean a class of combinatorially equivalent, simplicial complexes covering the same space, each complex being a formal manifold, meaning that the complement7 of each vertex is combinatorially equivalent to A or to A n-, according as the vertex in question is inside Mn or on Mn, where Ak stands for a closed k-simplex and Mn is the boundary of Mn. These covering complexes will be called proper triangulations of Mn (of course any simplicial complex covering Mn is a proper triangulation if the 'Hauptvermutung' is true). The proper triangulations of an unbounded manifold of class C', or smooth manifold, are to be Cl-triangulations.7a By a smooth, bounded, n-dimensional manifold we shall mean the manifold of which a sub-complex Ko C Kn is a proper triangulation, where Kn is a C' triangulation of a smooth, unbounded n-dimensional manifold and Ko is a formal manifold. By the topological product Mn X A k we shall mean the manifold having a normal subdivision of the cell-complex K n X A k as a proper triangulation where Kn is a proper triangulation of Mn. We shall use =to indicate combinatorial equivalence, and Mn Mn will mean that K Kn , where Ki is a proper triangulation of Mn .