Publication: Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces.
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1999
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American Mathematical Society
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Abstract
In this paper the authors review some work relating the topology to algebraic invariants. Three cases are considered: hypersurfaces, vector fields and vector fields tangent to hypersurfaces.
An example is the case of "real Milnor fibres". Let f R :B R →R be a real analytic function which extends to a function f on the closed unit ball B in C n. Assume that 0 is the only critical point of f and denote the real hypersurface f −1 R (δ) by V R + for small positive real δ .
Let A denote A/(f 0 ,⋯,f n ) , the quotient of the ring of germs of real analytic functions by the partial derivatives of f . Choose any R-linear map L:A→R which sends the Hessian of f to a positive number; then one has a non-degenerate bilinear form ⟨ , ⟩:A×A→ . A→ L R , for which the signature σ gives χ(V R + )=1−σ .
The authors use a result of D. Eisenbud and H. I. Levine [Ann. Math. (2) 106 (1977), no. 1, 19–44, to calculate the signature of the bilinear form; this is notoriously difficult to compute in practice.
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Proceedings of the 1st Iberoamerican "Cruz del Sur" Congress on Geometry held in Olmué, Chile, January 5–11, 1998