The Derivatives of Homotopy Theory

Johnson, Brenda

doi:10.2307/2154811

Cited by 18 publications

(26 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To identify D n I one has to know the action of the full group Σ n . Johnson [11] gave an explicit finite complex with Σ n -action whose S-dual is the answer. Arone and Mahowald [4] gave a different answer of that kind, showed that it was equivalent to Johnson's, and used it to make some very interesting calculations.…”

Section: This Pattern Persists For Any N ≥ 1 An N Th Degree Homogenementioning

confidence: 99%

See 1 more Smart Citation

Calculus III: Taylor Series

2003

View full text Add to dashboard Cite

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n − 1)-excisive part, can be classified: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen's algebraic K -theory.

show abstract

Section: This Pattern Persists For Any N ≥ 1 An N Th Degree Homogenementioning

confidence: 99%

“…Johnson [11] defined a based finite complex K n with Σ n -action whose S -dual is ∂ (n) I( * ). Her K n was designed to admit an interesting map…”

Section: Remarkmentioning

confidence: 99%

Calculus III: Taylor Series

2003

View full text Add to dashboard Cite

show abstract

“…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…”

Section: )mentioning

confidence: 99%

Quadratic homology

Baues

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We describe axioms for a 'quadratic homology theory' which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group. IntroductionThe development of algebraic topology was profoundly affected by the notion of homology. Originally homology had coefficients in abelian groups and EilenbergSteenrod described the axioms of such an ordinary homology theory. Somewhat later important examples of generalized homology theories were found which led to the notion of homology with coefficients in a spectrum. The spectrum-homology can also be described as a homology theory satisfying all Eilenberg-Steenrod axioms except the dimension axiom concerning the value of the homology on spheres. In fact this value characterizes a homology theory in the sense that a natural transformation of homology theories which is an isomorphism on spheres is also an isomorphism on all finite CW-complexes.In his recent work on the calculus of homotopy functors Goodwillie observed that a spectrum is equivalent to a linear homotopy functor and spectrum-homology of a space X can be equivalently described by the homotopy groups.where L is a linear homotopy functor. We therefore call spectrum-homology also a linear homology theory. The non-linear homology theories are then obtained by the homotopy groupswhere D is any homotopy functor. Hence we again generalize homology by choosing now homotopy functors as coefficients. Clearly this is a very far reaching generalization since in particular homotopy groups of a space are the homology groups with coefficients in the identity functor. It is an old problem to find axioms which characterize the theory of homotopy groups in a similar way as ordinary homology theory is characterized by the Eilenberg-Steenrod axioms. The identity functor is

show abstract

“…(here we may use strict orbits instead of homotopy orbits, because the action of Σ m+1 on U(m) is free). Recalling from [Jo95,AM97] that C m+1 (the m + 1-th Goodwillie derivative of the identity) is given…”

Section: Weiss' Calculusmentioning

confidence: 99%

“…The references for this material are [G90, G92, G3]. See also [Jo95] for an exposition of some of the material in [G3]. Let D n (X) be the n-th layer in the Goodwillie tower of the identity.…”

Section: Introductionmentioning

confidence: 99%

Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Arone

1998

Mathematical Research Letters

View full text Add to dashboard Cite

Abstract. We study certain cross-effects of the unstable homotopy of spheres. These cross-effects were constructed by Weiss, for different purposes, in the context of "Orthogonal calculus". We show that Mithchell's finite spectra with A kfree cohomology (constructed in [Mt85]) arise naturally as stabilizations of Weiss' cross-effects. In particular, we find that after a suitable Bousfield localization, our cross-effects, which capture meaningful information about the unstable homotopy of spheres, are homotopy equivalent to the infinite loop spaces associated with Mitchell's spectra. This last result is a partial generalization of the main result of Mahowald and the author in [AM97]. IntroductionLet X be a topological space. We are interested in studying the difference between the homotopy type of X and that of S 2 X, the double suspension of X (for technical reasons the statements turn out to be a little cleaner if one works with double suspensions rather than single suspensions). One naive way to compare X and S 2 X would be by means of the map X → S 2 X given by taking the smash product of the identity map on X with the inclusion S 0 → S 2 . Of course, this map is null-homotopic and thus is unlikely to provide useful information about the difference between X and S 2 X. A much better idea is to consider the Freudental (double) suspension map w 1 : X → Ω 2 S 2 X. Let F 1 (X) be the homotopy fiber of this map. We regard F 1 (X) as measuring the difference between the homotopy type of X and that of S 2 X. We would like to iterate the process and find higher analogues of the suspension map and higher iterated differences of the suspension map. Thus at the second stage we would like to find a suitable way to compare F 1 (X) with F 1 (S 2 X). Obviously, there exists a natural map F 1 (X) → Ω 2 F 1 (S 2 X). However, this map turns out to be null-homotopic, just as the map X → S 2 X is null-homotopic. It follows from the 1991 Mathematics Subject Classification. 55P40, 55P42, 55P65.

show abstract

The Derivatives of Homotopy Theory

Abstract: 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.

Cited by 18 publications

References 0 publications

Calculus III: Taylor Series

Calculus III: Taylor Series

Quadratic homology

Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology

Contact Info

Product

Resources

About