RT Book, Section
T1 Computation of topological numbers via linear algebra: hypersurfaces, vector fields and vector fields on hypersurfaces.
A1 Giraldo Suárez, Luis
A1 Gómez-Mont, X.
A1 Mardešic, P.
A2 Carocca, Ángel
A2 González-Aguilera, Víctor
A2 Rodríguez, Rubí E.
AB 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.
PB American Mathematical Society
SN 0-8218-1381-1
YR 1999
FD 1999
LK https://hdl.handle.net/20.500.14352/60710
UL https://hdl.handle.net/20.500.14352/60710
NO Proceedings of the 1st Iberoamerican "Cruz del Sur" Congress on Geometry held in Olmué, Chile, January 5–11, 1998
DS Docta Complutense
RD 7 abr 2025