K. Lonsted's Algebraic Geometry PDF

By K. Lonsted

ISBN-10: 3540095276

ISBN-13: 9783540095279

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4 (Multiplicativity of the signature). Let M and N be closed oriented manifolds, then sign(M × N ) = sign(M ) · sign(N ). Proof. This is a standard application of the K¨ unneth Theorem and the algebraic fact from above, that the signature of an unimodular symmetric bilinear form over the rational numbers vanishes if there is a submodule of half rank on which the form is identically zero. The K¨ unneth theorem allows one to decompose the middle cohomology of M × N as the tensor product of the middle cohomology groups of M and N plus the orthogonal sum given by tensor products of the other terms.

J∗ ([∂W ]) α) ¯ ∪ j ∗ (β, = 0, 20 Chapter 3. The Signature Thus the intersection form vanishes on im(j ∗ ). By Poincar´e duality the intersection form S(∂W ) ⊗ Q is non-degenerate. Since the dimension of im(j ∗ ) is 1/2 · dim(H k (∂W )), the proof is finished by the considerations above from linear algebra. 3 Multiplicativity and other Properties of the Signature The signature of a disjoint union is the sum of the signatures and so we obtain a homomorphism sign : Ωm → Z. Here we define the signature as zero if the dimension of the manifold is not divisible by 4.

Then T OP is the union of T OP (n) which we identify via f × id with a subgroup of T OP (n + 1), and B(T OP ) is its classifying space. Actually Brumfiel considers B(P L) instead of B(T OP ), but for n > 1 the groups π4n (BP L) and π4n (B T OP ) are isomorphic [130]. Brumfiel shows that for n > 1 there is no map f : π4n (BT OP ) → π4n (BO) with f i∗ = id, where i : BO → BT OP is induced from the inclusion BO → BT OP . This follows from his main result in [37]. 30 Chapter 4. The Signature Theorem and the Novikov Conjecture Combined with the following elementary result we obtain the desired information.

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Algebraic Geometry by K. Lonsted

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