Special spines of piecewise linear manifolds

S. V. Matveev

Sbornik. Mathematics, Tome 21 (1973) no. 2, pp. 279-291

Résumé

A class of so-called special polyhedra is defined for each $n>1$. The following theorems are proved: 1. Every piecewise linear manifold $M^{n+1}$ with boundary can be collapsed to some $n$-dimensional special polyhedron. 2. The manifold $M^{n+1}$ is uniquely determined by this special polyhedron. 3. If $n\geqslant3$, then any special polyhedron can be thickened to an $(n+1)$-dimensional manifold. The author also gives applications of the results obtained to a series of questions connected with the Zeeman conjecture about the collapsibility of $P^2\times I$, where $P^2$ is a contractible polyhedron. Figures: 4. Bibliography: 6 titles.

Bibliographie

[1] B. G. Casler, “An embedding theorem for connected 3-manifolds with boundary”, Proc. Amer. Math. Soc., 16 (1966), 559–566 | DOI | MR

[2] J. W. Alexander, “On the subdivision of 3-space by a polyhedron”, Proc. Nat. Acad. Sci. USA, 10 (1924), 6–8 | DOI

[3] E. C. Zeeman, Seminar on combinatorial topology, (mimeographed), Inst. Hautes Études. Sci., Paris, 1963

[4] E. M. Brown, “The hauptvermutung for 3-complexes”, Trans. Amer. Math. Soc., 144 (1969), 173–196 | DOI | MR | Zbl

[5] P. Dierker, “Note on collapsing $K\times I$ where $K$ is a contractible polyhedron”, Proc. Amer. Math. Soc., 19:2 (1968), 425–428 | DOI | MR | Zbl

[6] C. P. Rourke, B. T. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, Berlin–Heidelberg–New York, 1972 | MR | Zbl