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Lecture 11: Hirzebruch’s signature theorem

In this lecture we define the signature of a closed oriented n-manifold for n divisible by four. It is a bordism invariant Sign: Ω n → Z. (Recall that we defined a Z/2Z-valued bordism invariant of non-oriented manifolds in Lecture 2.) The signature is a complete bordism invariant of closed oriented 4-manifolds (see (2.28)), as we prove here. It can be determined by tensoring with Q, or even tensoring with R. We use the general Pontrjagin-Thom Theorem 10.33 to convert the computation of this invariant to a homotopy theory problem. We state the theorem that all such bordism invariants can be determined on products of complex projective spaces. In this lecture we illustrate the techniques necessary to compute that Ω 4 ⊗ Q is a one-dimensional rational vector space. The general proof will be sketched in the next lecture. Here we also prove Hirzebruch’s formula assuming the general result. We sometimes tensor with R instead of tensoring with Q. Tensoring with R has the advantage that real cohomology is represented by differential forms. Also, the computation of the real cohomology of BSO can be related to invariant polynomials on the orthogonal Lie algebra so.