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Homology via embedded simplexes: a new proof of Lalonde’s theorem. (Homologie des simplexes plongés: Une preuve nouvelle du théorème de Lalonde.) (French) Zbl 0713.57014
Let V be a smooth n-dimensional differentiable manifold. For an integer k $$(1\leq k<\infty)$$, one denotes by $$S^ k(V)$$ the subcomplex of the singular chain complex S(V) generated in all dimensions $$p\geq 0$$ by the simplexes which are the $$C^ k$$-differentiable maps $$\Delta^ p\to V$$, where $$\Delta^ p$$ is the standard p-dimensional simplex. Fix k and denote $$S^ k(V)$$ 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 $$S^ P(V)$$. The homology groups of $$S^ P(V)$$ are denoted by $$H_ p^ P(V)$$ (p$$\geq 0)$$ and are called the homology groups of the embedded simplexes of V. For $$p>n$$ the equality $$H_ p^ P(V)=0$$ is true. Since V can be endowed with a differentiable triangulation [J. H. C. Whitehead, Ann. Math., II. Ser. 41, 809-824 (1940; Zbl 0025.09203)], the natural morphism $$H_ p^ P(V)\to H_ p(V)$$ is surjective for $$0\leq p\leq n$$. In 1987, 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 $$H_ p^ P(V)\to H_ p(V)$$ is bijective for $$0\leq p\leq n-1$$. The author gives a simpler, more combinatorial proof of F. Lalonde’s theorem.
Reviewer: I.Pop

##### MSC:
 57R19 Algebraic topology on manifolds and differential topology 55N10 Singular homology and cohomology theory
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##### References:
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