By H. Blaine Lawson
This e-book bargains a scientific and finished presentation of the ideas of a spin manifold, spinor fields, Dirac operators, and A-genera, which, over the past twenty years, have come to play an important position in lots of parts of recent arithmetic. because the deeper purposes of those principles require a number of common different types of the Atiyah-Singer Index Theorem, the theorems and their proofs, including all prerequisite fabric, are tested right here intimately. The exposition is richly embroidered with examples and functions to a large spectrum of difficulties in differential geometry, topology, and mathematical physics. The authors continuously use Clifford algebras and their representations during this exposition. Clifford multiplication and Dirac operator identities are even utilized in position of the traditional tensor calculus. This special approach unifies the entire general elliptic operators in geometry and brings clean insights into curvature calculations. the basic relationships of Clifford modules to such themes because the concept of Lie teams, K-theory, KR-theory, and Bott Periodicity additionally obtain cautious attention. a different function of this publication is the advance of the speculation of Cl-linear elliptic operators and the linked index theorem, which connects definite sophisticated spin-corbordism invariants to classical questions in geometry and has ended in probably the most profound kin identified among the curvature and topology of manifolds.
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Extra info for Spin geometry
Example text
Et ::::: (v /\ { te1eh12 ... e,pP '" (v /\ - V L)q> if il • V L)q> • tfll te 1 for =1 > 1. 22) holds on an additive basis of Cl", it holds in general. _ §4. HI (= quatemions). With little difficulty the eader can check the first few cases: Cf. 1 •0 = C Cl o•1 = R EB R Cl 2 •0 Cf. 1 = R(2) vhere R(2) denotes the algebra of 2 x 2 real matrices. 1. 2 '" Cf. " ® Cf. 2) Cl"s ® Cl 1 ,1 '" Cl,+ t,s+ 1 or all n,r,s > O. \,fote that here we are using the ungraded tensor product. 3) 26 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proof.
We begin with a general definition. Let V be a vector space over a field k and let q be a quadratic form on V. 1. Let K ~ k be a field containing k. Then a K -representation of the Clifford algebra Ct(V,q) is a k-algebra homomorphism DEFINITION p: ct(V,q) ---+ HomK(W, W) into the algebra of linear transformations of a finite dimensional vector space W over K. The space W is called a ct(V,q)-module over K. We shall often simplify notation by writing p(q»(w) =q> . 1) for q> e Ct(V,q) and we W, when no confusion is likely to occur.
17), we have that (~(x),x) == (v,;X,x) == O. • , v,. are pointwise linearly independent. Fix x E SN and consider the linear map i:1I:: R" -+ TxS N c: RN+ 1 given by ix(v) = v' x The image of ix is the linear span of JI1 (x), ... (x), so it suffices to prove that ix is injective. However if ixv = v . x = 0, then v . V' x = -llvl1 2 x = 0 and so v = O. Since V;( - x) = - Jj(x), these vector fields descend to (pointwise linearly independent) vector fields on pN(A). • The question now is: given an integer N, what is the largest number of independent vector fields on SN that can be constructed in this manner?