Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Uniqueness of product and coproduct decompositions in rational homotopy theory


Authors: Roy Douglas and Lex Renner
Journal: Trans. Amer. Math. Soc. 264 (1981), 165-180
MSC: Primary 55P62
DOI: https://doi.org/10.1090/S0002-9947-1981-0597874-3
MathSciNet review: 597874
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a nilpotent rational homotopy type such that (1) $ S(X)$, the image of the Hurewicz map has finite total rank, and (2) the basepoint map of $ M$, a minimal algebra for $ X$, is an element of the Zariski closure of $ {\text{Aut}}(M)$ in $ {\text{End}}(M)$ (i.e. $ X$ has "positive weight"). Then (A) any retract of $ X$ satisfies the two properties above, (B) any two irreducible product decompositions of $ X$ are equivalent, and (C) any two irreducible coproduct decompositions of $ X$ are equivalent.


References [Enhancements On Off] (What's this?)

  • [1] R. Body and R. Douglas, Tensor products of graded algebras and unique factorization, Amer. J. Math. 101 (1979), 909-914. MR 536045 (80h:16002)
  • [2] -, Rational homotopy and unique factorization, Pacific J. Math. 68 (1977).
  • [3] -, Unique factorization of rational homotopy types, Pacific J. Math. (to appear). MR 599316 (82d:55009)
  • [4] R. Body and D. Sullivan, Zariski dynamics of a homotopy type (unpublished notes).
  • [5] A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 0251042 (40:4273)
  • [6] A. Bousfield and V. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., No. 179 (1976). MR 0425956 (54:13906)
  • [7] A. Bousfield and D. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, New York, 1972. MR 0365573 (51:1825)
  • [8] A. Deschner, Sullivan's theory of minimal models, Thesis, Univ. of British Columbia, 1976.
  • [9] S. Halperin, Lecture notes on minimal models, Université des Sciences et Techniques de Lille, 1977.
  • [10] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent spaces and groups, North-Holland Math. Studies, no. 15, North-Holland, Amsterdam, 1975.
  • [11] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal imbeddings. I, Lecture Notes in Math., vol. 339, Springer-Verlag, New York, 1973. MR 0335518 (49:299)
  • [12] M. Mimura and H. Toda, On $ p$-equivalences and $ p$-universal spaces, Comment. Math. Helv. 4 (1971), 87-97. MR 0285007 (44:2231)
  • [13] J. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137-204. MR 516917 (80e:55020)
  • [14] L. Renner, Automorphism groups of minimal algebras, Thesis, Univ. of British Columbia, 1978.
  • [15] A. Stone, Inverse limits of compact spaces, General Topology Appl. 10 (1979), 204-211. MR 527845 (80m:54014)
  • [16] D. Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1-79. MR 0442930 (56:1305)
  • [17] -, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977) 269-332. MR 0646078 (58:31119)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55P62

Retrieve articles in all journals with MSC: 55P62


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0597874-3
Keywords: Minimal algebra, rational homotopy, $ I$-split category, idempotent, positive weight, algebraic group, toroidal imbedding
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society