A -Theoretical Proof of Hartogs Extension Theorem on Stein by Ruppenthal J.

By Ruppenthal J.

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Examples of immersed, but not embedded, submanifolds. 7 Submanifolds 33 6-shaped submanifold is not constant. Note that the curve intersects itself at t = −1 and t = 2, but because t = 2 is not part of the domain, one says that the curve touches itself at the origin (0, 0). 47. 9(b)). It is not injective. However, it is injective when restricted to, say, the range −1 < t < ∞. 48. A more striking example of a self-touching submanifold is given by the image of the mapping f : R → R2 so that f (t) = ( 1t , sin πt) (0, t + 2) for 1 ≤ t < ∞, for − ∞ < t ≤ −1.

Informally speaking, attaching an n-cell to a CW-complex is carried out by identifying the boundary of the cell with the finite union of a subset of (n−1)cells in the complex. Therefore, by using this attachment rule, and starting off with the empty set, a CW-complex X can be inductively constructed out by gluing the 0-cells, 1-cells, 2-cells, and so forth; this originates a filtration X (−1) ⊆ X (0) ⊆ X (1) ⊆ X (2) ⊆ · · · of X such that X = i≥−1 X (i) , with X (−1) = ∅. The set X (i) obtained from X (i−1) by attaching the collection of i-cells is nothing more than the i-skeleton of X.

To say that f : M → N is an immersion means that the differential D f (p) is injective at every point p ∈ M . This is the same as saying that the Jacobian matrix of f has rank equal to dim M (which is only possible if dim M ≤ dim N ). 42. Let M , N be two manifolds of dimensions m, n, respectively, and f : M → N a smooth mapping. The mapping f is an immersion if and only if for each point p ∈ M there are coordinate systems (U, ϕ), (V, ψ) about p and f (p), respectively, such that the composite ψ f ϕ−1 is a restriction of the coordinate inclusion ι : Rm → Rm × Rn−m .

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