Twisted free tensor products
HTML articles powered by AMS MathViewer
- by Elyahu Katz PDF
- Trans. Amer. Math. Soc. 257 (1980), 91-103 Request permission
Abstract:
A twisted free tensor product of a differential algebra and a free differential algebra is introduced. This complex is proved to be chain homotopy equivalent to the complex associated with a twisted free product of a simplicial group and a free simplicial group. In this way we turn a geometric situation into an algebraic one, i.e. for the cofibration $Y \to Y { \cup _g} CX \to \Sigma X$ we obtain a spectral sequence converging into $H(\Omega (Y { \cup _g} CX))$. The spectral sequence obtained in the above situation is similar to the one obtained by L. Smith for a cofibration. However, the one we obtain has more information in the sense that differentials can be traced, requires more lax connectivity conditions and does not need the ring of coefficients to be a field.References
- Israel Berstein, On co-groups in the category of graded algebras, Trans. Amer. Math. Soc. 115 (1965), 257–269. MR 206941, DOI 10.1090/S0002-9947-1965-0206941-6
- Edgar H. Brown Jr., Twisted tensor products. I, Ann. of Math. (2) 69 (1959), 223–246. MR 105687, DOI 10.2307/1970101
- V. K. A. M. Gugenheim, On a theorem of E. H. Brown, Illinois J. Math. 4 (1960), 292–311. MR 112135
- Daniel M. Kan, On monoids and their dual, Bol. Soc. Mat. Mexicana (2) 3 (1958), 52–61. MR 111035
- Elyahu Katz, Twisted Cartesian and free products, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 60 (1976), no. 3, 235–239 (English, with Italian summary). MR 464234
- Elyahu Katz, A Künneth formula for coproducts of simplicial groups, Proc. Amer. Math. Soc. 61 (1976), no. 1, 117–121 (1977). MR 445502, DOI 10.1090/S0002-9939-1976-0445502-3
- J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
- Norbert H. Schlomiuk, Principal cofibrations in the category of simplicial groups, Trans. Amer. Math. Soc. 146 (1969), 151–165. MR 258039, DOI 10.1090/S0002-9947-1969-0258039-2
- Larry Smith, Lectures on the Eilenberg-Moore spectral sequence, Lecture Notes in Mathematics, Vol. 134, Springer-Verlag, Berlin-New York, 1970. MR 0275435
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 257 (1980), 91-103
- MSC: Primary 55R99; Secondary 55U10
- DOI: https://doi.org/10.1090/S0002-9947-1980-0549156-2
- MathSciNet review: 549156