Resumen de Cohomologías elípticas (Un ensayo introductorio)

Guillermo Moreno

• Let a and ß be any angles then the known formula sin (a+ß) = sina cosß + cosa sinß becomes under the substitution x = sina, y = sinß, sin (a + ß) = x v(1 - y2) + y v(1 - x2) =: F(x,y). This addition formula is an example of "Formal group law", which show up in many contexts in Modern Mathematics.

  In algebraic topology suitable cohomology theories induce a Formal group Law, the elliptic cohomologies are the ones who realize the Euler addition formula (1778): F(x,y) =: (x vR(y) + y vR(x)/1 - ex2y2). For R(z) = 1 - 2dz2 + ez4 the above case corresponds to e=0, d=1/2.

  In this survey paper we define these cohomology theories and establish their relationship with global analysis (Atiyah-Singer theorem) and modular forms following ideas of Landweber, Hirzebruch et al.