# ITERATED LOOP FUNCTORS AND THE HOMOLOGY OF THE STEENROD ALGEBRA A(P)

#### Abstract

Let A be the mod p Steenrod algebra and let M(,A) be the category of unstable A-modules and degree-preserving A-maps. This paper begins on a program for the calculation of the homology groups H('k)(M) which arise as the E(,2)-terms of unstable Adams spectral sequence.^ Let (OMEGA): M(,A) (--->) M(,A) be the "loop" functor on the category of unstable A-modules mod-p. Roughly, (OMEGA)M is obtained by lowering the dimensions in M by 1, and dividing out by the smallest submodule necessary to assure the result is unstable. Let (OMEGA)('k): M(,A) (--->) M(,A) be the k-fold iterate of (OMEGA). The author poses the problem: compute the left derived functors^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ Two considerations suggest this problem. The first is that the derived functors of (OMEGA)('k) are related to the derived functors of (OMEGA)('k-1) by a simple short exact sequence:^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ So that an inductive process to solve the problem seems possible. The second is that a complete solution to this problem would include as a special case a description of the homology group H('k)(M). In fact, one has that H('k)(M) is the component of (OMEGA)('k)M is degree zero.^ Let S('n) = H*(S('n),Z(,p)). This paper determines completely the unstable A-modules (OMEGA)('k)S('n) for the cases s = 0,1,2 in which k (LESSTHEQ) n+s-1. In the case s = 1, the author introduces an algebra over A called L(,1). As an algebra, L(,1) = Z(,p){u(,0)} (CRTIMES) E{v(,0)}, a tensor product of a polynomial algebra and an exterior algebra. The author gives explicit formulas for A-action on u(,0), v(,0), and defines certain sub-quotients F('k)L('n) of L(,1). Then he proves^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ as A-modules for all k (LESSTHEQ) n, where (SUMM)('a) is a-fold iterate of the suspension functor (SUMM). The case s = 2 is somewhat more complicated. Thus the author starts with a tensor product Z(,p){(sigma)(,1),(sigma)(,2)} (CRTIMES) E{(rho)(,1),(rho)(,2)} of polynomial algebra and exterior algebra, which is equipped with A-action satisfying Cartan formula. Then he localizes this algebra by inverting the element (sigma)(,2). An A-algebra L(,2) is then obtained as a sub-object of this localization; in fact, L(,2) = Z(,p){(sigma)(,1),(sigma)(,2)} (CRTIMES) E{(rho)(,1),(rho)(,2)} (CRPLUS) Span^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ The author defines sub-quotients F('k)L('n) of L(,2) and proves^ (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)^ as A-modules for all k (LESSTHEQ) n+l. ^

#### Subject Area

Mathematics

#### Recommended Citation

LI, HU-HSIUNG, "ITERATED LOOP FUNCTORS AND THE HOMOLOGY OF THE STEENROD ALGEBRA A(P)" (1980). *ETD Collection for Fordham University*. AAI8020070.

https://fordham.bepress.com/dissertations/AAI8020070