Ivancevic_Applied-Diff-Geom

666 Applied Differential Geometry: A Modern IntroductionLet α ∈ H 1 (M) be such that α([φ t (x)]) = 1. So ∂α = 1. Then, for everyβ ∈ Ker∂, ∂(α∪β) = β. As ∂◦∂ = 0, this means that [Lalonde et al. (1998);Lalonde et al. (1999); Lalonde and McDuff (2002)]Moreover, the mapKer∂ ⊂ Im∂ =⇒ Ker∂ = Im∂.α∪ : H k (M) → H k+1 (M)is injective on Ker ∂ and H ∗ (M) decomposes as the direct sum Ker ∂ ⊕(α∪Ker ∂).The above corollary claims that for a symplectic loop φ,Ker ∂ = Im ∂ iff [φ t (x)] ≠ 0 in H 1 (M).Since 1 ∈ H 0 (M) is in Ker ∂ it must equal ∂(α) for some α ∈ H 1 (M). Thismeans that α([φ t (x)]) ≠ 0 so that [φ t (x)] ≠ 0. Note that the only placethat the symplectic condition enters in the above proof is in the claim that∂ ◦ ∂ = 0. Since this is always true when the loop comes from a circleaction, this lemma holds for all, not necessarily symplectic, circle actions.In this case, we can interpret the result topologically. For the hypothesis[φ t (x)] ≠ 0 in H 1 (X) implies that the action has no fixed points, so thatthe quotient M/S 1 is an orbifold with cohomology isomorphic to ker ∂.Thus, the argument shows that M has the same cohomology as the product(M/S 1 ) × S 1 .4.13 Clifford Algebras, Spinors and Penrose Twistors4.13.1 Clifford Algebras and ModulesIn this subsection, mainly following [Yang (1995)], we provide the generaltheory of Clifford algebra and subsequently consider its special 4D case.Let V be a vector space over R with a quadratic form Q on it. The Cliffordalgebra of (V, Q), denoted by C(V, Q), is the algebra over R generatedby V with the relationsv 1 · v 2 + v 2 · v 1 = −2Q(v 1 , v 2 ), (for all v 1 , v 2 ∈ V ).Since Q is symmetric, we have v 2 = −Q(v) for all v ∈ V .For fixed Q, we may abbreviate C(V, Q) and Q(v 1 , v 2 ) into C(V ) and(v 1 , v 2 ) respectively.