Abstract
A manifold is a space which is locally like euclidean space. Some of the most important topological spaces are manifolds: Lie groups and their homogeneous spaces are manifolds. If a (compact) Lie group operates on a manifold then the orbit of every point is a manifold; if the operation is sufficiently regular then the orbit space is also a manifold. The set of solutions x=(x 1, ... , x n )∈ℝn of a sufficiently regular system of equations α μ (x 1, ... , x n )=0, μ=1, ... , m, is a manifold. These and other examples justify studying the special homology properties of manifolds.
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Notes
- 1.
Proof. Clearly \( \bar{V} \subset \left( {V \cup X} \right) \) . Assume \( x \not\subset \bar{V} \); then X contains a small open (n−1)-ball D such that \( D \cap \bar{V} = \emptyset \). It follows that W∪D is open in ℝn, and ℝn−(X−D)=V∪(W∪D); in particular, V is a bounded component of ℝn−(X−D). But H n−1(X−D; ℤ2)≅Γ(X, D; ℤ2)=0, by 3.3; hence ℝn-(X−D) has no bounded component, by 3.7.
- 2.
We use d! from § 11 here, not 10.5, because we’ll apply 11.14.
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© 1995 Springer-Verlag Berlin Heidelberg
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Dold, A. (1995). Manifolds. In: Lectures on Algebraic Topology. Classics in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-67821-9_8
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DOI: https://doi.org/10.1007/978-3-642-67821-9_8
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