Franklin P. Peterson

A spectrum whose Zp cohomology is the algebra of reduced pth powers

Throughout this paper all cohomology groups will have 2, coefficients unless otherwise stated. All spectra will be O-connected. We will make various constructions on spectra, for example, forming fibrations and Postnikov systems, just as one does with topological spaces. For the details of this see [6]. If one wishes, one may read “spectrum” as iv-connected topological space, iV a large integer, add N to all dimensions in sight, and read all theorems as applicable in dimensions less than 2N.

A -generators for certain polynomial algebras

R. M. W. Wood has proved a conjecture we made about dimensions of A generators of H *(RP ∞ × … × RP ∞ ). In this note we give some of the consequences of this conjecture.

Spherical classes and the Dickson algebra

We attack the conjecture that the only spherical classes in the homology of Q 0 S 0 are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the E 2 -term of the unstable Adams spectral sequence converging to π∗( Q 0 S 0 ), using results about the Dickson algebra and by studying the Lannes–Zarati homomorphism.

The Kervaire invariant of

1. Statements of results. Let Bm(Spin), Qm(SU), and Qm{e) denote the wth spinor, special unitary, and framed cobordism groups respectively (see [6]). In [4] Kervaire defined a homomorphism

\left( {8k + 2} \right)

: Ö2w(e) —>Z2 for n odd and ttj^l, 3, or 7, and showed that

-manifolds

= 0 for w = 5. Kervaire and Milnor state in [5] that

GENERALIZED COHOMOTOPY GROUPS.

= 0 for n = 9. One of the corollaries of our results is that

Modules over the Steenrod algebra

= 0 for w = 4fe + l, ferlin [2] a homomorphism \F: 3>2n(Spin)—»Z2 was defined for In = 8&+2, k*zl, such that <I> = SEp, where p: 02n(^)—»Q2n(Spin) is the obvious map. \F induces a map from S22n(<S£/) into Z2 which we also denote by ^ . I t is easily verified that Qi(SU) = Oi(Spin) = Qi(e) =Z 2 . Let a be the generator. Let 6 be the secondary cohomology operation coming from the relation SqSq = 0 on an integer cohomology class [7]. If ƒ is a map, let 6/ denote the associated functional cohomology operation [7]. The main theorems of this announcement are the following.

A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras

computation of ir(K;X), the set of homotopy classes of maps of a complex K into a space X. When K is an n-sphere Sn, then 7r (K; X) =7r,, (X), the familiar n-th homotopy group of X. When X = Sn and the dimension of K is ? 2n-2, then ir(K; X) -7rsn(K), the n-th cohomotopy group of K. The structure of 7rn(K) has been studied in [11]. In this paper, we shall introduce cohomotopy groups with coefficients in an abelian group G; namely, 7rn (K; G) = 7r(K;X), where the dimension of K is II induces a unique

On immersions of n-Manifolds

In this paper we generalize this theorem to the modp Steenrod algebra. To do this there are several problems to be overcome. Firstly, a correct theorem must be formulated. This is done by finding elements a, in the Steenrod algebra with (ai)” = 0 and defining H(fif, ai) = Ker ai/Im(ai)P-‘. Secondly, the proof given in [l] does not seem to generalize to the mod p Steenrod algebra and hence we had to find a proof that did. Finally, the reduced power operations only generate a subalgebra of the modp Streenrod algebra and we had to extend our results from this subalgebra to the full Steenrod algebra.