Geodesic
Homology reduction of cycles in the~complement of an~algebraic hypersurface
Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1417-1427

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An example of a 3-dimensional cycle in the complement of an algebraic hypersurface $V\subset\mathbb C^3$ that cannot be deformed into a tube over (is not homologous to the coboundary of) a 2-dimensional cycle in the set of regular points of $V$ is presented. Thus, the corresponding result of Poincare in $\mathbb C^2$ fails in $\mathbb C^n$ for $n>2$. It is proved that Poincare's result holds for hypersurfaces in $\mathbb C^n$ with a 'thin' set of singularities that are complete intersections. 
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     author = {N. A. Buruchenko and A. K. Tsikh},
     title = {Homology reduction of cycles in the~complement of an~algebraic hypersurface},
     journal = {Sbornik. Mathematics},
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     number = {10},
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     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/}
}
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N. A. Buruchenko; A. K. Tsikh. Homology reduction of cycles in the~complement of an~algebraic hypersurface. Sbornik. Mathematics, Tome 186 (1995) no. 10, pp. 1417-1427. http://geodesic.mathdoc.fr/item/SM_1995_186_10_a1/