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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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