Cyclic Sullivan-de Rham forms

Author:
Christopher Allday

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3971-3982

MSC:
Primary 55N91; Secondary 18G60, 55P62

DOI:
https://doi.org/10.1090/S0002-9947-1995-1316843-9

MathSciNet review:
1316843

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a simplicial set the Sullivan-de Rham forms are defined to be the simplicial morphisms from to a simplicial rational commutative graded differential algebra (cgda). However is a cyclic cgda in a standard way. And so, when is a cyclic set, one has a cgda of cyclic morphisms from to . It is shown here that the homology of this cgda is naturally isomorphic to the rational cohomology of the orbit space of the geometric realization with its standard circle action. In addition, a cyclic cgda is introduced; and it is shown that the homology of the cgda of cyclic morphisms from to is naturally isomorphic to the rational equivariant (Borel construction) cohomology of .

**[A, P]**C. Allday and V. Puppe,*Cohomological methods in transformation groups*, Cambridge Studies in Advanced Mathematics 32, Cambridge Univ. Press, Cambridge, 1993. MR**1236839 (94g:55009)****[B, H, M]**M. Bökstedt, W.-C. Hsiang and I. Madsen,*The cyclotomic trace and algebraic**-theory of spaces*, Invent. Math.**111**(1993), 465-539. MR**1202133 (94g:55011)****[B, G]**A. K. Bousfield and V. K. A. M. Gugenheim,*On PL de Rham Theory and rational homotopy type*, Mem. Amer. Math. Soc.**179**(1976). MR**0425956 (54:13906)****[D, H, K]**W. G. Dwyer, M. J. Hopkins and D. M. Kan,*The homotopy theory of cyclic sets*, Trans. Amer. Math. Soc.**291**(1985), 281-289. MR**797060 (86m:55014)****[H]**S. Halperin,*Lectures on minimal models*, Mém. Soc. Math, France (N.S.), No. 9-10 (1983). MR**736299 (85i:55009)****[J]**J. D. S. Jones,*Cyclic homology and equivariant homology*, Invent. Math.**87**(1987), 403-423. MR**870737 (88f:18016)****[L]**J.-L. Loday,*Cyclic homology*, Grundlehren. Math. Wiss. 301, Springer-Verlag, Berlin, Heidelberg, 1992. MR**1217970 (94a:19004)****[M]**J. P. May,*Simplicial objects in algebraic topology*, Van Nostrand Math. Studies, no. 11, Van Nostrand, Princeton, NJ, 1967. MR**0222892 (36:5942)****[Sp]**J. Spaliński,*Strong homotopy theory of cyclic sets*, J. Pure Appl. Algebra**99**(1995), 35-52. MR**1325168 (96f:55012)****[S]**D. Sullivan,*Infinitesimal computations in topology*, Publ. Math. I.H.E.S.**47**(1977), 269-332. MR**0646078 (58:31119)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55N91,
18G60,
55P62

Retrieve articles in all journals with MSC: 55N91, 18G60, 55P62

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1316843-9

Article copyright:
© Copyright 1995
American Mathematical Society