Page 113 - Curvature and Homology

3.3. DERIVATIONS IN A GRADED ALGEBRA

for n = 2m and

for n = 2m + 1. Finally, from

we obtain

for n = 2m and

for n = 2m + 1 from which for n = 2m and 6 > 3 or n = 2m + 1
and 6 > 2(m - 1)/(8m - 5)

This completes the proof.
The following statement is immediately clear from theorem 3.2.1
and PoincarC duality:

Corollary. A 5-dintettssbnal! 8-pPSnched compact and orientable Riemannian
manifold is a homology sphere for 6 > 211 1.
The even dimensional case of the theorem should be compared with
theorem 6.4.1.

3.3. Derivations in a graded algebra

The tensor algebra of contravariant (covariant) tensors and the
Grassman algebra of differential forms are examples of a type of algebraic
structure known as a graded algebra. A graded algebra A over a field K
is defined by prescribing a set of vector spaces AP (p = 0, 1, a*.) over K
such that the vector space A is the direct sum of the spaces AP; further-