We will study the angle sums of polytopes, listed in the α-vector,
working to exploit the analogy between the f-vector of faces in each dimension
and the alpha-vector of angle sums. The Gram and Perles relations on the
α-vector are analogous to the Euler and Dehn-Sommerville relations on
the f-vector. First we describe the spaces spanned by the the alpha-vector and
the α-f-vectors of certain classes of polytopes. Families of polytopes
are constructed whose angle sums span the spaces of polytopes defined by the
Gram and Perles equations. This shows that the dimension of the affine span of
the space of angle sums of simplices is floor[(d-1)/2], and that of the
combined angle sums and face numbers of simplicial polytopes and general
polytopes are d-1 and 2d-3, respectively. Next we consider angle sums of
polytopal complexes. We define the angle characteristic on the alpha-vector in
analogy to the Euler characteristic. We show that the changes in the two
correspond and that, in the case of certain odd-dimensional polytopal
complexes, the angle characteristic is half the Euler characteristic. Finally,
we consider spherical and hyperbolic polytopes and polytopal complexes.
Spherical and hyperbolic analogs of the Gram relation and a spherical analog of
the Perles relation are known, and we show the hyperbolic analog of the Perles
relations in a number of cases. Proving this relation for simplices of
dimension greater than 3 would finish the proof of this result. Also, we show
how constructions on spherical and hyperbolic polytopes lead to corresponding
changes in the angle characteristic and Euler characteristic.Comment: Ph.D. Dissertatio