The virtual Spivak fiber, duality on fibrations and Gorenstein spaces

Murillo, Aniceto

doi:10.1090/S0002-9947-07-04048-2

The virtual Spivak fiber, duality on fibrations and Gorenstein spaces
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Trans. Amer. Math. Soc. 359 (2007), 3577-3587 Request permission

Abstract:

In this paper we study a generalization of the homology of the Spivak fiber of a $1$-connected space over any field and deduce consequences concerning Poincaré complexes, Gorenstein spaces and finiteness properties on fibrations.

References

Luchezar Avramov and Stephen Halperin, Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra, algebraic topology and their interactions (Stockholm, 1983) Lecture Notes in Math., vol. 1183, Springer, Berlin, 1986, pp. 1–27. MR 846435, DOI 10.1007/BFb0075446
Luchezar L. Avramov, Hans-Bjørn Foxby, and Jack Lescot, Bass series of local ring homomorphisms of finite flat dimension, Trans. Amer. Math. Soc. 335 (1993), no. 2, 497–523. MR 1068924, DOI 10.1090/S0002-9947-1993-1068924-3
Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
Nicolas Dupont and Kathryn Hess, Twisted tensor models for fibrations, J. Pure Appl. Algebra 91 (1994), no. 1-3, 109–120. MR 1255924, DOI 10.1016/0022-4049(94)90136-8
Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Gorenstein spaces, Adv. in Math. 71 (1988), no. 1, 92–112. MR 960364, DOI 10.1016/0001-8708(88)90067-9
Yves Félix, Steve Halperin, and Jean-Claude Thomas, Differential graded algebras in topology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 829–865. MR 1361901, DOI 10.1016/B978-044481779-2/50017-1
Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
Yves Félix, Jean-Michel Lemaire, and Jean-Claude Thomas, L’application d’évaluation, les groupes de Gottlieb duaux et les cellules terminales, J. Pure Appl. Algebra 91 (1994), no. 1-3, 143–164 (French, with English and French summaries). MR 1255927, DOI 10.1016/0022-4049(94)90139-2
Yves Félix and Aniceto Murillo, Gorenstein graded algebras and the evaluation map, Canad. Math. Bull. 41 (1998), no. 1, 28–32. MR 1618931, DOI 10.4153/CMB-1998-006-2
Michael Weiss, Embeddings from the point of view of immersion theory. I, Geom. Topol. 3 (1999), 67–101. MR 1694812, DOI 10.2140/gt.1999.3.67
Daniel Henry Gottlieb, Poincaré duality and fibrations, Proc. Amer. Math. Soc. 76 (1979), no. 1, 148–150. MR 534407, DOI 10.1090/S0002-9939-1979-0534407-8
H. Gammelin, Gorenstein space with nonzero evaluation map, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3433–3440. MR 1458300, DOI 10.1090/S0002-9947-99-02092-9
Kathryn P. Hess, Twisted tensor products of DGAs and the Adams-Hilton model for the total space of a fibration, Adams Memorial Symposium on Algebraic Topology, 1 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 175, Cambridge Univ. Press, Cambridge, 1992, pp. 29–51. MR 1170569, DOI 10.1017/CBO9780511526305.004
John R. Klein, Poincaré duality embeddings and fiberwise homotopy theory, Topology 38 (1999), no. 3, 597–620. MR 1670412, DOI 10.1016/S0040-9383(98)00034-2
John R. Klein, The dualizing spectrum of a topological group, Math. Ann. 319 (2001), no. 3, 421–456. MR 1819876, DOI 10.1007/PL00004441
Aniceto Murillo, Rational fibrations in differential homological algebra, Trans. Amer. Math. Soc. 332 (1992), no. 1, 79–91. MR 1079055, DOI 10.1090/S0002-9947-1992-1079055-X
Aniceto Murillo, On the evaluation map, Trans. Amer. Math. Soc. 339 (1993), no. 2, 611–622. MR 1112376, DOI 10.1090/S0002-9947-1993-1112376-1
Aniceto Murillo, The behaviour of $\textrm {Ext}_A(R,A)$ through a morphism, J. Algebra 192 (1997), no. 2, 477–503. MR 1452675, DOI 10.1006/jabr.1996.6953
E. H. Spanier, Homology theory of fiber bundles, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R.I., 1952, pp. 390–396. MR 0044838
Michael Spivak, Spaces satisfying Poincaré duality, Topology 6 (1967), 77–101. MR 214071, DOI 10.1016/0040-9383(67)90016-X