Constructing manifolds by homotopy equivalences I. An obstruction to constructing PL-manifolds from homology manifolds

Sato, Hajime

doi:10.5802/aif.406

Sato, Hajime

Annales de l'Institut Fourier, Tome 22 (1972) no. 1, pp. 271-286.

Résumé
Abstract

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 $H_{n - 4} (M; ℋ^{3})$ , 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 $M$ .

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 $H_{n - 4} (M, ℋ^{3})$ , 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 $M$ .

MR Zbl

DOI : 10.5802/aif.406

@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},
     pages = {271--286},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {22},
     number = {1},
     year = {1972},
     doi = {10.5802/aif.406},
     mrnumber = {49 #1522},
     zbl = {0219.57009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.406/}
}

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

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