On the Cobar Construction

On the Non-Existence of Elements of Hopf Invariant One

“…(iii) If n > 2, the elements {gflJhi} for which j # i + 1 are linearly independent in E2, 1 The same is true for n = 2, provided we exclude also the elements {g2,ihi+1}.…”

“…For example, we have said that 1 is "associated with" the relation Sq2Sq2 + Sq3Sq1 = 0. We would expect the composite operation Sq 3q to be "associated with" the relation ( 1 ) (Sq3Sq2)Sq2 + (Sq3Sq3)Sq1 = 0.…”

“…Our axioms will ensure that if an operation 1) is associated with such a pair (d, z), then it is defined on classes x in H1(X) such that 8Ck(x) = 0 for each k, and has values in H1 +q(X)/AKaH1 +qrk(X) where rk = deg (ak). (Note that a relation of degree (q + 1) corresponds to an operation of degree q.…”

“…Our next step is to choose a sequence of classes vz e Hl+q(E,) so that vA = v and so that the class vW+lu in Hl+q(f7E,1+) corresponds to v1 in Hl+q(E,) under the finite chain of equivalences. This choice is clearly possible and unique, because the suspension a Hr + (El + 1) Hr(&2Ez+ 1) is an isomorphism for r < 2(1 + 4). Let V>(for 1 > -) be the operation given by the universal example (El, {el}, vl).…”

The twisted Cartesian model for the double path fibration

“…We construct the generating category P and show how to lift the diagonal on the permutahedra P constructed above to a diagonal on Z. Naturally occurring examples of permutahedral sets include the double cobar construction, i.e., Adams' cobar construction [1] on the cobar with coassociative coproduct [3], [8], [13]. Permutahedral sets are distinguished from simplicial or cubical sets by their higher order structure relations.…”