The derivatives of homotopy theory

Abstract

We construct a functor of spaces, M" , and show that its multilinearization is equivalent to the nth layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type.

Growth of homotopy groups of spheres via the Goodwillie-EHP Sequence

“…We require the following lemma on connectivity of P 2 k (S n ), which is noted by Johnson [Joh95]. It follows immediately from the proof of Theorem 1.13 in [Goo03], using the fact that the identity functor is 1-analytic.…”

The partition poset complex and the Goodwillie derivatives of the identity in spaces

“…The equivalences from (2) show that ∂ 1 Id ≃ S. Johnson [Joh95] and later Arone-Mahowald [AM99] later gave a description of the higher homogeneous layers and derivatives of Id in terms of the partition poset complex.…”

“…Outline of the argument. We make use of the models for ∂ * Id described by Johnson [Joh95] and Arone-Mahowald [AM99], specifically, that the derivative of the identity can be identified with the totalization of the cobar complex on the commutative cooperad of spectra S: ∂ * Id ≃ holim ∆ Cobar(I, S, I).…”

Untitled

“…We know by (4.7) that quadratic homotopy groups π Q n form a quadratic homology theory on space 2 satisfying all properties in section 3. The cross effect of π Q n (X) is given by a natural isomorphism [19], [24]. Arone-Mahowald [1] mentioned that P 2 (X) also can be constructed by the fiber of the stable James-Hopf map γ 2 in [13]; that is…”