H ∗ (MO 〈8 〉; Z/2) is an Extended A ∗ 2 -Coalgebra

Abstract

We show that H*(MO(8>; Z/2) is an extended A*-coalgebra, where A* is the subalgebra of the Steenrod algebra generated by (Sq', Sq2, Sq4). The method yields an analogous result for H*( M Spin; Z/2). Recently Don Davis conjectured [D] that H*(MO(8); Z/2) is an extended A*-coalgebra and discussed various consequences of such a result. We apply the method introduced in [P] to prove his conjecture. Let A* C A* be the subalgebra generated by (Sq', Sq2, Sq4). THEOREM A. H*MO (8) is an extended A *-coalgebra, i.e., there is an A *-coalgebra N such that H*MO (8) A* ?Al N as an A*-coalgebra. COROLLARY (BAHRI AND MAHOWALD [BMJ). A*//A2 is a direct summand in H*MO (8). THEOREM B. H*M Spin is an extended A,*-coalgebra. PROOF OF THEOREM A. Let p: BO 3], where a(m) is the number of ones in the dyadic expansion of m. H*BO(8) is a sub-Hopf algebra of H* BO, and hence by Borel's theorem is also polynomial [B]. Let p1 be the coalgebra primitive in H2, l BO (and, via the Thom isomorphism, in H21l MO). From the inductive formula for Newton polynomials, pj is indecomposable in H* MO. So we can consider the polynomial subalgebra P2 = Z/2 [ p8, P4 P, p2, * *. C H*MO. P~7/~r84 2 3 l H M p * The map QH'BO n-* QH'BO(8) of indecomposable quotients is an isomorphism if a(n 1) > 3; and thus so is the map PH, BO<(8) ' PH, BO of coalgebra primitives. So the generators of P2 lie in H*MO (8), and thus all of P2 does. The coaction 4 on P2 C H*MO(8) is known [BP]: ipj = p,_. Since A2 (A*2)* = A/(8, 24, ,.24...), the augmentation ideal P2 is clearly a subcomodule of H*MO<(8) over A2 (although not over A). Thus so is the ideal I generated by P2 in H*MO(8). Received by the editors February 3, 1982. 1980 Mlathematicts Subjec-t Classification. Primary 57R90; Secondary 55N22. 55R40. 55S 10. 'Supported in part by a grant from the NSF. (01983 American Mathematical Society 0002-9939/82/0(X)0-0443/$( 1.50