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A look back at Poincaré’s Analysis Situs. (English) Zbl 0851.01003

Greffe, Jean-Louis (ed.) et al., Henri Poincaré. Science et philosophie: congrès international, mai 14-18, 1994, Nancy, France. Berlin: Akademie Verlag. 251-258 (1996).
The study under review is based on the Topology seminar held by its author – K. S. Sarkaria – at the University of Punjab, India, between 1993-1994. The lines along with the seminar was developed involved understanding of Poincaré’s mathematical outlook, as discernible from his Analysis Situs, as well as the way in which Poincaré’s ideas may be found in the future development of mathematics. There follows a brief presentation of the main concepts to be met in Analysis situs (and in its five Compléments), namely: boundary operator, Betti numbers, homologies; periods of indefinite integrals and de Rham cohomology; intersection matrices and Poincaré’s duality in orientable closed manifolds; triangulability of differentiable manifolds; monodromy of integral linear PDEs on a manifold and definition of the fundamental group; homotopy classes of loops and a classification theorem for some affine 3-manifolds. Largely discussed is Poincaré’s first definition of homology, and also his 3-manifold theory.
For the entire collection see [Zbl 0838.00010].
Reviewer: C.Irimia (Iaşi)

MSC:

01A60 History of mathematics in the 20th century
57-03 History of manifolds and cell complexes

Biographic References:

Poincaré, H.
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