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Blowing up fixed points

Publicacions de la Secció de Matemàtiques
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  • Mathematics


Pub . Mat . UAB N° 25 Juny 1981 BLOWING UP FIXED POINTS Francisco Gómez Ruiz Facultad de Ciencias, Universidad de Santander and Secció de Matemátiques, Universitat Autónoma de Barcelona . Spain . Rebut el 15 de Febrer del 1981 This note shows how to use the technique of blowing up submanifolds to give easier proofs of some theorems concerning fixed point sets of toral actions on smooth manifolds . None of the results given here is new . The main theorem (theorem 3) as well as the applications (8), (9) and (10) are well known . Nevertheless I believe proofs presented in this note are not the usual ones . (1) Let us begin by describing the equivariant blowing up of invariant subma- nifo1ds . Let G be a compact Lie group acting smoothly on a smooth manifold M and let B be a G-stable closed smooth submanifold of M. Consider the induced action of G on the tangent bundle TM of M. It is given by a .v = (dTa )x (v) x E M, a E G, v E Tx(M) where T X (M) denotes the tangent space of M at x and Ta is the diffeomorphism of M defined by Ta (x) = a .x . The above action of G on TM restricts to actions of G on TM IB and TB, since B is G-invariant . Thus we have an induced action of G on the normal bundle, v :E n+ B, of B in M . Let P(v) :P(E) -> B be the projective bundle associated to v(its fi- bre over x E B is the projective space associated to the quotient TX(M)/TX(B)) The bundle P(v)'inherits an obvioús action from the action of G on v . We also consider the canonical line bundle, É P(E), on P(E) (its fibre over z E P(E) consists of al] vectors of z) . The above actions of G on P(E) and E induce an action of G on E such that the map a :E - E, given by a(z,v) =v, is G-equivariant . Observe that a is surjective and restricts to a diffeomorphism É-P(E) ~O' E-B (we identity the base B to its image under the zero cross-sec- tion) . U of B in M together with a G-equivariant diffeomorphism ! E-P U such that 82 It is not diff

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