By Solomon Lefschetz

Because the ebook of Lefschetz's Topology (Amer. Math. Soc. Colloquium guides, vol. 12, 1930; mentioned lower than as (L)) 3 significant advances have motivated algebraic topology: the improvement of an summary advanced self reliant of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the strategy of Cech for treating the homology idea of topological areas through structures of "nerves" every one of that is an summary advanced. the result of (L), very materially further to either through incorporation of next released paintings and by means of new theorems of the author's, are the following 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, exact presentation of definitions and proofs. Examples are sparingly given, in most cases of an easy type, which, as they don't partake of the scope of the corresponding textual content, can be intelligible to an basic pupil. yet this is often essentially a publication for the mature reader, within which he can locate the theorems of algebraic topology welded right into a logically coherent complete

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**Example text**

10), Xf,XE 3(U) . 19) If df is expressed in terms of coordinates by df = h i dx1 + ... + h"dx", then (2. 11) and (2. +af dx". , n - 1, which agrees with (2. 22) where deg w denotes the degree of co, and where f A T1 means the same as frl if f E S2° U. This mapping d is called the exterior derivative. The proof of the theorem, which depends on nothing more than combining the rules of multivariable calculus with those of exterior algebra, will be given a little later. Condition (2. 22) is tantamount to the equality of the mixed second partial derivatives, and could be called the Iteration Rule.

V) w in L (V -4 W) , and extend to all of V` 0 W by linearity. The tensor product of 1-forms co', w2, denoted co' 0 0)2, is the map which assigns to each y e U the bilinear map w' ®w2 (y) E (TYR") * ® (TYR") * . w' ®w2 (y) (4, c) = co' (y) (4) (02 (y) (c). Thus for vector fields X, Y, we may write: w' ®w2 (X, y) = (co' . X) (w2 - Y) E C`° (U). For example, dx ® xdy (eza xe;. The relationship with the exterior -, ay + yaz) = product is: w1 AO)2 = w1 ®0)2-w20 w1. i where the { h13 } are smooth functions on U.

L f g. 2. , without using the notion of a derivation), that for any vector fields X and Yon an open set U c R", [X. Y1 defined by (2. 5) is indeed a vector field. , x"). Show that the two second-derivative terms cancel out. 3. Verify the Jacobi identity (2. 6) for vector fields. Hint: Work in terms of derivations; don't differentiate anything! 4. Show that if X and Y are the vector fields on U = R3\ { 0} given below, then LXY = 0: X= -{x2+y2+z2}-3/2{xa +ya +za ax ay az }; Y = -ya +xa . -3 Hint: For brevity, take r = {x2 + y2 + z2 } 1 /2 and note that (xar y ax ay -3 y ax ) = 0.