Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Postnikov sections of formal and hyperformal spaces


Author: Samuel B. Smith
Journal: Proc. Amer. Math. Soc. 122 (1994), 893-903
MSC: Primary 55P62; Secondary 55S45
DOI: https://doi.org/10.1090/S0002-9939-1994-1204385-5
MathSciNet review: 1204385
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let X be a simply connected CW complex and $ {X_n}$ its nth Postnikov section. We prove that X is formal provided $ {H^q}({X_n},\mathbb{Q})$ is additively generated by decomposables for all q and n with $ q > n$. Recall from [4] that a space X is said to be hyperformal if its rational cohomology algebra is the quotient of a free graded algebra by an ideal generated by a regular sequence. Using the main result of Felix and Halperin's paper (Trans. Amer. Math. Soc. 270 (1982), 575-588) we show our sufficient condition for formality is actually equivalent to hyperformality.


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

  • [1] M. Arkowitz, Formal differential graded algebras and homomorphisms, J. Pure Appl. Algebra 51 (1988), 35-52. MR 941888 (89h:55021)
  • [2] A. K. Bousfield and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., no. 179, Amer. Math. Soc., Providence, RI, 1976. MR 0425956 (54:13906)
  • [3] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-275. MR 0382702 (52:3584)
  • [4] Y. Felix and S. Halperin, Formal spaces with finite dimensional rational homotopy, Trans. Amer. Math. Soc. 270 (1982), 575-588. MR 645331 (83h:55023)
  • [5] S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Adv. Math. 32 (1979), 233-279. MR 539532 (80j:55016)
  • [6] I. Kaplansky, Commutative rings, Univ. Chicago Press, Chicago, 1974. MR 0345945 (49:10674)
  • [7] J. Neisendorfer and T. Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), 565-580. MR 0500938 (58:18429)
  • [8] J. Oprea, DGA homology decompositions and a condition for formality, Illinois J. Math. 30 (1986), 122-137. MR 822387 (87i:55024)
  • [9] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331. MR 0646078 (58:31119)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55P62, 55S45

Retrieve articles in all journals with MSC: 55P62, 55S45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1204385-5
Keywords: Rational homotopy theory, formality, hyperformality, minimal model
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society