Abstract
In the first paper on L’Analysis Situs, dated 1895, Poincaré introduced fundamental notions which are nowadays called differential manifolds, complexes, Betti numbers, fundamental groups, etc., thus laying down the foundations of modern algebraic topology. In addition Poincaré posed the problem of determining the Betti numbers of differential manifolds by means of exterior differential forms; see Section 9 of that paper. The problem was clarified by E. Cartan, and only in 1931 was it completely solved by de Rham. The result, now known as the de Rham theorem, may be stated as follows.
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References
P. Baum and L. Smith, The real cohomology of differentiable fiber bundles. Comm. Math. Hely. 42, 171 - 179 (1967).
R. Bott, Lectures on characteristic classes and foliations. In Lectures on Algebraic and Differential Topology( R. Bott, S. Gitler, and I. M. James, Eds.), Springer, Berlin, Heidelberg, New York, 1972.
E. Friedlander, P. A. Griffiths, and J. Morgan, Homotopy theory and differential forms. Mimeo. Notes, 1972.
H. Poincaré, Analysis Situs. t. 1 J. de L’Ecole Polytechnique, 1895.
D. Sullivan, Differential forms and topology of manifolds. Symp. Tokyo on Topology, 1973.
Infinitesimal Computations in Topology. Publ. Math. No. 47, Inst. de Hautes Etudes Scientifiques, 1978.
A. Weil, Sur les theoremes de de Rham. Comm. Math. Hely. 20, 119 - 145 (1952).
J. Wolf, The real and rational cohomology of differential fiber bundles. Trans. Amer. Math. Soc. 245, 211 - 220 (1975).
Wu Wen-tsün, Theory of /*-functions in algebraic topology—Real topology of homogeneous space. (In Chinese) Acta Math, Sinica 18, 162 - 172 (1975).
Real topology of fiber squares. Scientia Sinica 18, 464 - 482 (1975).
Effective calculation and axiomatization of 1*-functors on complexes. Scientia Sinica 10, 647 - 664 (1976).
Wu Wen-tsün and Wang Qi-ming, I*-functor of a fibre space. Scientia Sinica 21, 1 - 18 (1978).
Wu Wen-tsün, Calculability of I* -Measure vis-à-vis Space-Union and Related Constructions. Kuxue Tong Pao, 1979.
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Wen-tsün, W. (1980). de Rham-Sullivan Measure of Spaces and Its Calculability. In: Hsiang, WY., Kobayashi, S., Singer, I.M., Wolf, J., Wu, HH., Weinstein, A. (eds) The Chern Symposium 1979. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8109-9_10
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