![]() Thom then allows us to define Stiefel–Whitney classes if the Poincaré duality algebra has the structure of an unstable algebra over the Steenrod algebra. We use one of Milnor’s Theorems to define an analogous element for any Poincaré duality algebra over a field and show that it is a Macaulay dual for the kernel of the multiplication map of that algebra. Here M should be a closed smooth oriented manifold of dimension n. As defined there it is an element of the relative cohomology algebra $$H^n(M \times M ,M \times M \setminus \Delta (M))$$ H n ( M × M, M × M \ Δ ( M ) ) where $$\Delta : M \mathop \limits M \times M$$ Δ : M ↪ M × M is the diagonal embedding and provides a replacement for the Thom class (see e.g., and page 56 et seq) of the tangent bundle of M. ![]() Milnor calls the diagonal cohomology class in Chapter 11 and is used by him in his discussion of Poincaré duality and characteristic classes of manifolds. The purpose of this note is to provide a purely algebraic interpretation for the cohomology class J.
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