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Zbl 0713.57014
Cerf, J.
Homology via embedded simplexes: a new proof of Lalonde's theorem. (Homologie des simplexes plongés: Une preuve nouvelle du théorème de Lalonde.) (French)
[J] Bull. Soc. Math. Fr. 118, No. 1, 1-25 (1990). ISSN 0037-9484

Let V be a smooth n-dimensional differentiable manifold. For an integer k $(1\le k<\infty)$, one denotes by $S\sp k(V)$ the subcomplex of the singular chain complex S(V) generated in all dimensions $p\ge 0$ by the simplexes which are the $C\sp k$-differentiable maps $\Delta\sp p\to V$, where $\Delta\sp p$ is the standard p-dimensional simplex. Fix k and denote $S\sp k(V)$ by S(V). The set of the simplexes of S(V) which are embeddings is stable for all face operators and it defines in S(V) a chain subcomplex (without degeneration operators) denoted by $S\sp P(V)$. The homology groups of $S\sp P(V)$ are denoted by $H\sb p\sp P(V)$ (p$\ge 0)$ and are called the homology groups of the embedded simplexes of V. For $p>n$ the equality $H\sb p\sp P(V)=0$ is true. Since V can be endowed with a differentiable triangulation [{\it J. H. C. Whitehead}, Ann. Math., II. Ser. 41, 809-824 (1940; Zbl 0025.09203)], the natural morphism $H\sb p\sp P(V)\to H\sb p(V)$ is surjective for $0\le p\le n$. In 1987, {\it F. Lalonde} [Mem. Soc. Math. Fr., Nouv. Sér. 30 (1987; Zbl 0642.57019)] proved the following J. H. C. Whitehead's conjecture: The natural morphism $H\sb p\sp P(V)\to H\sb p(V)$ is bijective for $0\le p\le n-1$. The author gives a simpler, more combinatorial proof of F. Lalonde's theorem.
[I.Pop]

MSC 2000:: *57R19 Algebraic topology on manifolds
55N10 Singular theory

Citations: Zbl 0025.09203; Zbl 0642.57019

Cited in: Zbl 0911.57019

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