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 Mat. Sb. (N.S.), 1979, Volume 110(152), Number 2(10), Pages 278–296 (Mi msb2450)

The fundamental groups of manifolds and Poincaré complexes

V. G. Turaev

Abstract: In this article the fundamental groups of $n$-dimensional manifolds and $n$-dimensional Poincaré; complexes with $[n/2]$-connected universal coverings are studied. Special attention is given to the case $n=3$: it is established that the fundamental groups of closed three-dimensional manifolds possess dual presentations in a certain sense, and purely algebraic conditions are found that are necessary and sufficient for a given group to be isomorphic to the fundamental group of some Poincaré; complex of formal dimension three. With the help of these conditions the symmetry of the Alexander invariants of finite Poincaré; complexes of formal dimension three is established. In the case $n\ne3$ analogous results are proved (the presentations of a group by generators and relations are replaced by segments of resolutions of the fundamental ideal of a group ring, and the Alexander invariants are replaced by their generalizations).
Figures: 1.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Sbornik, 1981, 38:2, 255–270

Bibliographic databases:

UDC: 513.86
MSC: Primary 57M05; Secondary 57P10

Citation: V. G. Turaev, “The fundamental groups of manifolds and Poincaré complexes”, Mat. Sb. (N.S.), 110(152):2(10) (1979), 278–296; Math. USSR-Sb., 38:2 (1981), 255–270

Citation in format AMSBIB
\Bibitem{Tur79}
\by V.~G.~Turaev
\paper The fundamental groups of manifolds and Poincar\'e complexes
\jour Mat. Sb. (N.S.)
\yr 1979
\vol 110(152)
\issue 2(10)
\pages 278--296
\mathnet{http://mi.mathnet.ru/msb2450}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=552116}
\zmath{https://zbmath.org/?q=an:0449.57001|0414.57001}
\transl
\jour Math. USSR-Sb.
\yr 1981
\vol 38
\issue 2
\pages 255--270
\crossref{https://doi.org/10.1070/SM1981v038n02ABEH001327}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Turaev V., “3-Dimensional Poincaré Complexes - Classification and Splitting”, 257, no. 3, 1981, 551–552
2. V. G. Turaev, “Three-dimensional Poincaré complexes: homotopy classification and splitting”, Math. USSR-Sb., 67:1 (1990), 261–282
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