AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Goodwillie tower and the EHP sequence
About this Title
Mark Behrens, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 218, Number 1026
ISBNs: 978-0-8218-6902-4 (print); 978-0-8218-9014-1 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00645-3
Published electronically: November 21, 2011
Keywords: Unstable homotopy groups of spheres,
Goodwillie calculus,
EHP sequence,
Dyer-Lashof operations
MSC: Primary 55Q40; Secondary 55Q15, 55Q25, 55S12
Table of Contents
Chapters
- Introduction
- 1. Dyer-Lashof operations and the identity functor
- 2. The Goodwillie tower of the EHP sequence
- 3. Goodwillie filtration and the $P$ map
- 4. Goodwillie differentials and Hopf invariants
- 5. EHPSS differentials
- 6. Calculations in the $2$-primary Toda range
- A. Transfinite spectral sequences associated to towers
Abstract
We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime $2$. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to recompute the $2$-primary unstable stems through the Toda range (up to the $19$-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching’s operad structure on the derivatives of the identity. These operations act on the mod $2$ stable homology of the Goodwillie layers of any functor from spaces to spaces.- Greg Arone and Michael Ching, Operads and chain rules for calculus of functors, to appear in Astérisque.
- G. Z. Arone and W. G. Dwyer, Partition complexes, Tits buildings and symmetric products, Proc. London Math. Soc. (3) 82 (2001), no. 1, 229–256. MR 1794263, DOI 10.1112/S0024611500012715
- Gregory Z. Arone, William G. Dwyer, and Kathryn Lesh, Loop structures in Taylor towers, Algebr. Geom. Topol. 8 (2008), no. 1, 173–210. MR 2377281, DOI 10.2140/agt.2008.8.173
- Greg Arone and Marja Kankaanrinta, A functorial model for iterated Snaith splitting with applications to calculus of functors, Stable and unstable homotopy (Toronto, ON, 1996) Fields Inst. Commun., vol. 19, Amer. Math. Soc., Providence, RI, 1998, pp. 1–30. MR 1622334, DOI 10.1016/s0022-4049(97)00050-9
- Gregory Z. Arone and Kathryn Lesh, Augmented $\Gamma$-spaces, the stable rank filtration, and a $bu$ analogue of the Whitehead conjecture, Fund. Math. 207 (2010), no. 1, 29–70. MR 2576278, DOI 10.4064/fm207-1-3
- Greg Arone and Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3, 743–788. MR 1669268, DOI 10.1007/s002220050300
- Greg Arone, Iterates of the suspension map and Mitchell’s finite spectra with $A_k$-free cohomology, Math. Res. Lett. 5 (1998), no. 4, 485–496. MR 1653316, DOI 10.4310/MRL.1998.v5.n4.a6
- Mark Behrens, The Goodwillie tower for ${S}^1$ and Kuhn’s theorem, Available at $\mathtt {http://arxiv.org/abs/1012.0810}$.
- A. K. Bousfield and D. M. Kan, Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 (1971), 1006–1010. MR 296935, DOI 10.1090/S0002-9904-1971-12837-9
- R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132
- Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005), 833–933. MR 2140994, DOI 10.2140/gt.2005.9.833
- F. R. Cohen, J. P. May, and L. R. Taylor, Splitting of certain spaces $CX$, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 465–496. MR 503007, DOI 10.1017/S0305004100055298
- Thomas G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory, $K$-Theory 4 (1990), no. 1, 1–27. MR 1076523, DOI 10.1007/BF00534191
- Thomas G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711. MR 2026544, DOI 10.2140/gt.2003.7.645
- Thomas G. Goodwillie, Calculus. II. Analytic functors, $K$-Theory 5 (1991/92), no. 4, 295–332. MR 1162445, DOI 10.1007/BF00535644
- Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
- Po Hu, Transfinite spectral sequences, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 197–216. MR 1718081, DOI 10.1090/conm/239/03602
- I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107
- Brenda Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1295–1321. MR 1297532, DOI 10.1090/S0002-9947-1995-1297532-6
- N. J. Kuhn, S. A. Mitchell, and S. B. Priddy, The Whitehead conjecture and splitting $B(\textbf {Z}/2)^{k}$, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 255–258. MR 656206, DOI 10.1090/S0273-0979-1982-15027-3
- Nicholas J. Kuhn, The geometry of the James-Hopf maps, Pacific J. Math. 102 (1982), no. 2, 397–412. MR 686560
- Nicholas J. Kuhn, A Kahn-Priddy sequence and a conjecture of G. W. Whitehead, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 3, 467–483. MR 677471, DOI 10.1017/S0305004100060175
- Nicholas J. Kuhn, Chevalley group theory and the transfer in the homology of symmetric groups, Topology 24 (1985), no. 3, 247–264. MR 815479, DOI 10.1016/0040-9383(85)90001-1
- Nicholas J. Kuhn, New relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 185–216. MR 1851255
- Mark Mahowald, The metastable homotopy of $S^{n}$, Memoirs of the American Mathematical Society, No. 72, American Mathematical Society, Providence, R.I., 1967. MR 0236923
- Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
- J. Peter May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR 0281196
- Mark E. Mahowald and Douglas C. Ravenel, The root invariant in homotopy theory, Topology 32 (1993), no. 4, 865–898. MR 1241877, DOI 10.1016/0040-9383(93)90055-Z
- Stewart B. Priddy, Transfer, symmetric groups, and stable homotopy theory, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 244–255. Lecture Notes in Math. Vol. 341. MR 0350727
- Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
- Shin-ichiro Takayasu, On stable summands of Thom spectra of $B(\textbf {Z}/2)^n$ associated to Steinberg modules, J. Math. Kyoto Univ. 39 (1999), no. 2, 377–398. MR 1709300, DOI 10.1215/kjm/1250517919
- Hirosi Toda, Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR 0143217
- Michael Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3743–3796. MR 1321590, DOI 10.1090/S0002-9947-1995-1321590-3
- George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508