String Theory and M-Theory

442 String geometry
to a zero form
∂A0
ddA0 = d dxµ =
∂x µ ∂2A0 ∂x µ ∂xν dxµ ∧ dx ν , (9.227)
which vanishes due to antisymmetry of the wedge product. A p-form is
called closed if
dAp = 0, (9.228)
and exact if there exists a globally defined (p − 1)-form Ap−1 such that
Ap = dAp−1. (9.229)
A closed p-form can always be written locally in the form dAp−1, but this
may not be possible globally. In other words, a closed form need not be
exact, though an exact form is always closed.
Let us denote the space of closed p-forms on M by C p (M) and the space
of exact p-forms on M by Z p (M). Then the pth de Rham cohomology group
H p (M) is defined to be the quotient space
H p (M) = C p (M)/Z p (M). (9.230)
H p (M) is the space of closed forms in which two forms which differ by an
exact form are considered to be equivalent. The dimension of H p (M) is
called the Betti number. Betti numbers are very basic topological invariants
characterizing a manifold. The Betti numbers of S 2 and T 2 are described in
Fig. 9.14. Another especially important topological invariant of a manifold
is the Euler characteristic, which can be expressed as an alternating sum of
Betti numbers
χ(M) =
d
(−1) i bi(M). (9.231)
i=0
The Betti numbers of a manifold also give the dimensions of the homology
groups, which are defined in a similar way to the cohomology groups. The
analog of the exterior derivative d is the boundary operator δ, which acts
on submanifolds of M. Thus, if N is a submanifold of M, then δN is its
boundary. This operator associates with every submanifold its boundary
with signs that take account of the orientation. The boundary operator is
also nilpotent, as the boundary of a boundary is zero. Therefore, it can
be used to define homology groups of M in the same way that the exterior
derivative was used to define cohomology groups of M. Arbitrary linear
combinations of submanifolds of dimension p are called p-chains. Here again,
to be more precise, one should say what type of coefficients is used to form