Motives over S

This chapter shows that the operation φ‎ 𝑉 of the definition introduced in the previous chapter extends to a cohomology operation over 𝑘, and that it satisfies the recognition criterion of a theorem, so that φ‎ 𝑉 must be β‎𝑃𝑏. This construction of the cohomology operation utilizes the machinery of motives over a simplicial noetherian scheme. The chapter first presents this scheme in three parts, initially summarizing the basic theory of motives over a scheme 𝑆 before discussing motives over a simplicial scheme and over a smooth simplicial scheme. It then presents the slice filtration and generalizes from simplicial scheme 𝔛 to embedded schemes. Finally, this chapter defines the operations φ‎ 𝑖 and φ‎ 𝑉.

Motivic Classifying Spaces

This chapter focuses on motivic classifying spaces. It first connects the motives 𝑆∞ tr(𝕃𝑛) to cohomology operations on 𝐻2𝑛, 𝑛, at least when char(𝑘)=0. This parallels the Dold–Thom theorem in topology, which identifies the reduced homology ̃𝐻*(𝑋, ℤ) of a connected space 𝑋 with the homotopy groups of the infinite symmetric product 𝑆∞𝑋. A similar analysis shows that 𝔾𝑚 represents 𝐻1,1(−, ℤ), which allows us to describe operations on 𝐻1,1. The chapter then introduces the notion of scalar weight operations on 𝐻2𝑛, 𝑛. Afterward, it develops formulas for 𝑆𝓁tr(𝕃𝑛). These formulas imply that 𝑆∞ tr(𝕃𝑛) is a proper Tate motive, so there is a Künneth formula for them. The chapter culminates in a theorem demonstrating that β‎𝑃𝑏 is the unique cohomology operation of scalar weight 0 in its bidegree.

Frame Fields on Manifolds

Consider the following stable secondary cohomology operations associated with the relations in the mod 2 Steenrod algebra: such thatLet ψ5 be a stable tertiary cohomology operation associated with the above relation. We assume that (ϕ4, ϕ5) and ψ5 are chosen to be spin trivial in the sense of Theorem 3.7 of [14].Let ϕ0,0, ϕ1,1 be the stable Adams basic secondary cohomology operations associated with the relations:respectively.

Cohomology operations and duality

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operationwhich is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker productsif h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

On Künneth suspensions

In (2) we defined the Künneth suspension of a cohomology operation —the Künneth suspension involves an arbitrary css-complex Y rather than the 1-sphere S1, as with the usual suspension of a cohomology operation. Now the suspension homomorphism is well known to be related to the operation of forming loop spaces (cf. (4)). The main object of this paper is to prove a similar result for the Künneth suspension.