Odd primary resolutions

and classification of the stable summands

of stunted lens spaces

Author:
Jesús González

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1149-1169

MSC (1991):
Primary 55P15; Secondary 55N20, 55P42, 55Q50, 55T15.

Published electronically:
March 10, 1999

MathSciNet review:
1491866

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Abstract | References | Similar Articles | Additional Information

Abstract: The classification of the stable homotopy types of stunted lens spaces and their stable summands can be obtained by proving the triviality of certain stable classes in the homotopy groups of these spaces. This is achieved in the 2-primary case by Davis and Mahowald using classical Adams spectral sequence techniques. We obtain the odd primary analogue using the corresponding Adams spectral sequence based at the spectrum representing odd primary connective -theory. The methods allow us to answer a stronger problem: the determination of the smallest stunted space where such stable classes remain null homotopic. A technical problem prevents us from giving an answer in all situations; however, in a quantitative way, the number of cases missed is very small.

**1.**J. F. Adams,*Lectures on generalised cohomology*, Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three), Springer, Berlin, 1969, pp. 1–138. MR**0251716****2.**J. Hejtmanek,*Scattering theory of the linear Boltzmann-operator*, Methoden und Verfahren der mathematischen Physik, Band 9, Bibliographisches Institut, Mannheim, 1973, pp. 183–201. MR**0364905****3.**Donald M. Davis,*Odd primary 𝑏𝑜-resolutions and 𝐾-theory localization*, Illinois J. Math.**30**(1986), no. 1, 79–100. MR**822385****4.**Kh. N. Inasaridze,*Characterization of exact bifunctor homology theories*, Soobshch. Akad. Nauk Gruzin. SSR**121**(1986), no. 2, 241–243 (Russian, with English and Georgian summaries). MR**863395****5.**Donald M. Davis and Mark Mahowald,*The image of the stable 𝐽-homomorphism*, Topology**28**(1989), no. 1, 39–58. MR**991098**, 10.1016/0040-9383(89)90031-1**6.**S. Feder, S. Gitler, and K. Y. Lam,*Composition properties of projective homotopy classes*, Pacific J. Math.**68**(1977), no. 1, 47–61. MR**0515518****7.**J. González. The regular complex in the Adams spectral sequence.*Trans. Amer. Math. Soc.*350 (1998), 2629-2664. CMP**98:11****8.**J. González. A vanishing line in the Adams spectral sequence. To appear in*Topology*.**9.**Jesus Gonzalez,*Classification of the stable homotopy types of stunted lens spaces for an odd prime*, Pacific J. Math.**176**(1996), no. 2, 325–343. MR**1434994****10.**David Copeland Johnson and W. Stephen Wilson,*Projective dimension and Brown-Peterson homology*, Topology**12**(1973), 327–353. MR**0334257****11.**Richard M. Kane,*Operations in connective 𝐾-theory*, Mem. Amer. Math. Soc.**34**(1981), no. 254, vi+102. MR**634210**, 10.1090/memo/0254**12.**Wolfgang Lellmann,*Operations and co-operations in odd-primary connective 𝐾-theory*, J. London Math. Soc. (2)**29**(1984), no. 3, 562–576. MR**754942**, 10.1112/jlms/s2-29.3.562**13.**Wolfgang Lellmann and Mark Mahowald,*The 𝑏𝑜-Adams spectral sequence*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 593–623. MR**876468**, 10.1090/S0002-9947-1987-0876468-1**14.**Robert D. Thompson,*The 𝑣₁-periodic homotopy groups of an unstable sphere at odd primes*, Trans. Amer. Math. Soc.**319**(1990), no. 2, 535–559. MR**1010890**, 10.1090/S0002-9947-1990-1010890-8

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Additional Information

**Jesús González**

Affiliation:
Departamento de Matemáticas, Cinvestav, AP 14-740, México DF 07000

Email:
jesus@math.cinvestav.mx

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02284-9

Received by editor(s):
November 18, 1994

Received by editor(s) in revised form:
August 1, 1997

Published electronically:
March 10, 1999

Article copyright:
© Copyright 1999
American Mathematical Society