Differentiable Imbeddings
نویسندگان
چکیده
1. Terminology. V and M will be differentiable manifolds of dimension n and m respectively; differentiable meaning always of class C. For simplicity, we assume V compact and without boundary. We shall have to consider several categories of maps: (1) the category of continuous maps, (2) the category of topological imbeddings, (3) the category of topological immersions: a map ƒ: F—>M is a topological immersion of V in M if the restriction of ƒ to some neighborhood of each point of V is an imbedding, (4) the category of differentiable immersions: a m a p / : F—»Af belongs to this category if ƒ is differentiable of rank n = dim V everywhere, (5) the category of differentiable imbeddings: a differentiable imbedding ƒ : V-+M is a topological imbedding which is also a differentiable immersion. Two maps /o , / i : V-+M in one of the preceding categories are said to be homotopic in this category, if there exists a map F: VXR—+M (called a homotopy from ƒ<> to / i ) such that F | VX {O} =ƒ<,, ^ | VX {1} = / i and the associated map (x, t)-*(F(x, /), /) of VXR in MXR belongs to the given category. A homotopy in the category of differentiable imbeddings is also called a differentiable isotopy (cf. [4]).
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