Constructing manifolds by homotopy equivalences I. An obstruction to constructing PL-manifolds from homology manifolds
Annales de l'Institut Fourier, Volume 22 (1972) no. 1, p. 271-286

We aim at constructing a PL-manifold which is cellularly equivalent to a given homology manifold ${M}^{n}$. The main theorem says that there is a unique obstruction element in ${H}_{n-4}\left(M,{ℋ}^{3}\right)$, where ${ℋ}^{3}$ is the group of 3-dimensional PL-homology spheres modulo those which are the boundary of an acyclic PL-manifold. If the obstruction is zero and $M$ is compact, we obtain a PL-manifold which is simple homotopy equivalent to $M$.

Nous visons à construire une variété semi-linéaire qui est cellulairement équivalente à une variété homologique ${M}^{n}$ donnée. Le théorème dit qu’il y a un élément d’obstruction unique dans ${H}_{n-4}\left(M;{ℋ}^{3}\right)$, où ${ℋ}^{3}$ est un groupe de sphères homologiques qui sont des variétés semi-linéaires. Les éléments triviaux de ${ℋ}^{3}$ sont ceux qui sont un bord d’une variété semi-linéaire acyclique. Si l’obstruction est zéro et $M$ compacte, nous obtenons une variété semi-linéaire qui est simplement homotopiquement équivalente à $M$.

@article{AIF_1972__22_1_271_0,
author = {Sato, Hajime},
title = {Constructing manifolds by homotopy equivalences I. An obstruction to constructing PL-manifolds from homology manifolds},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {22},
number = {1},
year = {1972},
pages = {271-286},
doi = {10.5802/aif.406},
zbl = {0219.57009},
mrnumber = {49 \#1522},
language = {en},
url = {http://www.numdam.org/item/AIF_1972__22_1_271_0}
}

Sato, Hajime. Constructing manifolds by homotopy equivalences I. An obstruction to constructing PL-manifolds from homology manifolds. Annales de l'Institut Fourier, Volume 22 (1972) no. 1, pp. 271-286. doi : 10.5802/aif.406. http://www.numdam.org/item/AIF_1972__22_1_271_0/

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