Celebratio Mathematica — Atiyah

Papers on K-theory (1959–1970)

Works connected to Johan L. Dupont

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M. F. Atiyah and J. L. Dupont: “Vector fields with finite singularities,” Acta Math. 128 : 1 (1972), pp. 1–40. MR 0451256 Zbl 0233.57010 article

In this paper we give some generalizations of the famous theorem of H. Hopf which states that the number of singularities of a tangent vector field on a compact smooth manifold is equal to the Euler characteristic. Instead of a single vector field we consider \( r \) vector fields \( u_1,\dots,u_r \) and we are interested in their “singularities”, that is, the set \( \Sigma \) of points on the manifold at which they become linearly dependent. In general \( \Sigma \) will have dimension \( r-1 \), it is a cycle and its homology class is the \( (n-r+1) \)-th Stiefel–Whitney class of the manifold. This is the standard primary obstruction theory and it provides one way of generalizing the classical Hopf Theorem. However, this theory says nothing about \( \Sigma \) if \( \dim\Sigma < r-1 \). In this paper following E. Thomas [1967] we shall generalize the Hopf theorem by considering the other extreme case in which \( \Sigma \) is finite, so that \( \dim\Sigma = 0 \).