Page 79 - Curvature and Homology

2.2. SINGULAR HOMOLOGY 6 1
that if I KIP I and I Ki I are homeomorphic, the corresponding homology
groups of K and K' are isomorphic and their betti numbers coincide.
By a p-simplex [p, : Sp], p = 0, 1,2, on a differentiable manifold M
is understood an Euclidean p-simplex S* (point, closed line segment,
closed triangle, -..) together with a differentiable map g, of SP into M.
More precisely, let R* denote the vector space whose points are infinite
sequences of real numbers (xl, ..a, xn, ***) with only a finite number
of coordinates xn # 0. The finite-dimensional vector spaces Iip are
canonically imbedded in R". Consider the ordered sequence of points
(Po, -*-, Pp) (necessarily linearly independent) in R"D and denote by
P,)
A(Po,***, the smallest convex set containing them, that isA(Po,-**, Pp) =
@,,Po + + r,P, I r, 2 O,ro + + r, = 1). Let n(Po, ***, P,) =
(r; Po + + rb Pp 1 r; + -.* + 7; = 11, that is, the plane determined
by the P,, i = 0, *..,p. The numbers r;, -, r; are called bmycentk
coordinates of a vector in n(Po, -, P,). By a singular p-simplex on M
we mean a map p, of class 1 of A(Po, P,) into M. A singular p-chain
is a map of the set of all singular p-simplexes into R usually written as
a formal sum Zg& (gi E 2) with the singular simplexes indexed in
some fixed manner.
We denote by Is* I the support of s*, that is the set of pointsy(A(PO,-*-, P,)).
A chain is called locally finite if each compact set meets only a finite
number of supports with g, # 0. We consider only locally finite chains.
A singular chain is said to be finite if there are only a finite number of
non-vanishing g,. The support of a p-chain is the union of all I sr I with
gt # 0. Singular chains may be added and multiplied by scalars (elements
of R) in the obvious manner. Infinite sums are permissible if the result
is a locally finite chain.
The facu of a p-simplex sp [p,: Po,..., Pp] (p > 0) are the simplexes
4-1 = [p,: Po, P,-,, Pi+,, P,]. A boundary operator a is defined
by putting

For p = 0 we put as0 = 0. The extension to arbitrary singular chains is
by linearity. It is easily checked that the condition of local finiteness
is fulfilled. Moreover, aa = Q. Note that [sp: $-I] = (- I),.
Cycles and boundaries are defined in the usual manner. Let S, denote
the vector space of all finite p-chains, SC, the subspace of p-cycles and
Si the space of boundaries of finite (p + 1)-chains. The quotient
Sg/Si is called the pth singular homology space a group of M and is
denoted by SH,.
In this way, it is possible to associate with M a covering complex K,