By Solomon Lefschetz
Because the booklet of Lefschetz's Topology (Amer. Math. Soc. Colloquium courses, vol. 12, 1930; observed lower than as (L)) 3 significant advances have stimulated algebraic topology: the improvement of an summary complicated autonomous of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the tactic of Cech for treating the homology conception of topological areas by way of platforms of "nerves" every one of that is an summary complicated. the result of (L), very materially extra to either by way of incorporation of next released paintings and via new theorems of the author's, are right here thoroughly recast and unified when it comes to those new thoughts. A excessive measure of generality is postulated from the outset.
The summary standpoint with its concomitant formalism allows succinct, designated presentation of definitions and proofs. Examples are sparingly given, in general of an easy variety, which, as they don't partake of the scope of the corresponding textual content, will be intelligible to an common scholar. yet this can be essentially a booklet for the mature reader, during which he can locate the theorems of algebraic topology welded right into a logically coherent entire
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Extra resources for Algebraic Topology (Colloquium Publications, Volume 27)
Pontryagin proved this result by classifying framed I -manifolds in s m . With considerably more difficulty he was able to show that there are j ust two homotopy classes also in the case m = p + 2 � 4, using framed 2-manifolds . However, for m - p > 2 this approach to the problem runs into manifold diffi culties. It has since turned out to be easier to enumerate homotopy classes of mappings by quite different, more algebraic methods . * Pontryagin's construction is, however, a double-edged tool.
Computing the derivative of M, we see that w at a zero z t dWz (h) dWz (h) = dUz(h) for all = h for h h t TMz t TM� . Thus the determinant of dwz is equal to the determinant of dvz • Hence the index of w at the zero z is equal to the index L of v at z. Now according to Lemma 3 the index sum I: L is equal to the degree of g. This proves Theorem 1. EXAMPLE S . On the sphere 8m there exists a vector field v which points "north" at every point. * At the south pole the vectors radiate outward ; hence the index is + 1 .
Neighborhood of M * A different interpretation of this degree has been given by Allendoerfer and Fenchel : the degree of g can be expressed as the integral over M of a suitable curva ture scalar, thus yielding an m-dimensional version of the classical Gauss-Bonnet theorem. (References [ 1J, . See also Chern [6J . ) 39 The index sum We will also consider the squared distance function = cp (x) I l x - r (x) W· An easy computation shows that the gradient of 'I' is given by grad 'I' = 2(x - r(x» .
Algebraic Topology (Colloquium Publications, Volume 27) by Solomon Lefschetz