# 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://research.library.fordham.edu/dissertations/AAI8020070