On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}\mathrm{P}^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}\mathrm{P}^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}\mathrm{P}^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.

UNUSUAL FORMULAE FOR THE EULER CHARACTERISTIC

Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristic.

Regular neighbourhoods and mapping cylinders

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

Construction of Transverse Fields

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold,Mn, in Euclidean space, which are sufficient to prove thatMnhas a transverse field (see 1.1 and 1.2, definitions).In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifoldMn.Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).After the research for this paper was completed, a paper of Davis (2) came to our attention.

Structures on M × R

A central role in the theory of smoothing combinatorial manifolds is played by the Cairns–Hirsch Theorem, which may be expressed (in a weak form) as follows:If M is a combinatorial manifold and if M × R has a differentiable structure a, compatible with its combinatorial structure then M has a differentiable structure λ, such that (M × R)α is diffeomorphic with Mλ × R.