By J. L. Dupont, I. H. Madsen
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Extra resources for Algebraic Topology, Aarhus 1978
Here is the definition. 1. COORIENTATION AND LINKING INDEX 35 is transversal to l , then the intersection index is the algebraic number of points (taken with signs) in Mk whose image under it lies in I"'` . To get the sign at a point, look at the tangent space of M at the image of that point, and take the intersection of the cooriented subspace tangent to I'k with the oriented subspace that is the image of the tangent space of Mk under the differential of n . If it is not transversal to 1'k then replace it by any map ft that is transversal to 1 k and sufficiently close to 7r.
3. SEPARATING SOLUTIONS OF PFAFF EQUATIONS 41 Another sufficient condition is contained in Proposition 2 below. A point a E M belonging to the closure of U is said to be positive (negative) if f(x) tends to +oo (-x) when x E U tends to a. PROPOSITION 2. , f (x) tends to either +oc or to -oo when x tends to a boundary point in U), (2) the complement of U in M can be represented as the union of two nonintersecting closed sets, W, and W_ , such that WI does not contain negative points, and W, does not contain positive points.
Then the graph of y = f (x) , cooriented by the form d y - dx. separates Mx PROOF. We construct a spanning film for the graph. , the set of points (x, y) such that x E U and y = f (x) . We prove that the set M_ is the required spanning film. (I) We first prove that each point in M_ that does not belong to the graph of the function is an interior point of this set. , for points (a, y) where a E U n WI . But a sufficiently small neighbourhood in M of a E U n H does not contain points of W. and contains only points of U at which f has large values (as f (x) +oc when x tends in U to a).
Algebraic Topology, Aarhus 1978 by J. L. Dupont, I. H. Madsen