By James W. Vick
This booklet is designed to be an advent to a few of the fundamental rules within the box of algebraic topology. particularly, it truly is dedicated to the principles and purposes of homology thought. the single prerequisite for the scholar is a easy wisdom of abelian teams and aspect set topology. The necessities of singular homology are given within the first bankruptcy, in addition to probably the most vital functions. during this manner the scholar can quick see the significance of the fabric. The successive issues contain attaching areas, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology items, manifolds, Poincare duality, and glued aspect concept. in the course of the booklet the technique is as illustrative as attainable, with various examples and diagrams. Extremes of generality are sacrificed once they are inclined to vague the fundamental thoughts concerned. The publication is meant to be simply learn by way of scholars as a textbook for a direction or as a resource for person examine. the second one variation has been considerably revised. It encompasses a new bankruptcy on protecting areas as well as illuminating new workouts.
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Example text
Suppose n > 1 and f:D n+ Sn-' is a map such that f o i = identity, where i is the inclusion of Sn-' in D". 7 28 1 SINGULAR HOMOLOGY. THEORY homomorphisms is commutative: However, this gives a factorization of the identity on an infinite cyclic group through zero which is impossible. Therefore, no such retraction f exists. 18 COROLLARY (Brouwer $xed-point theorem) Given a map f: D" -+ Dn,there exists an x in D nwith f ( x ) = x . - PROOF Suppose f: Dn D" without fixed points. E Dn there is a well-defined ray starting at f ( x ) and passing through x .
If X is a topological space define - - - - - D = {(x,x)~ X E X }C X X X - the diagonal in X x X . Recall that X is Hausdorff if and only if the diagonal is a closed subset of X x X . Now let be an equivalence relation on X and denote by L3 the diagonal in ( X / - ) x (X/-). Note that the continuous function x x n: X x X + (X/-) x (X/-) has (nx n>-'(d) = ((x, Y ) 1 x Y } . - This subset of X x X is the graph of rhe relation. closed if and only if its graph is a closed subset of the above that if X / - is a Hausdorff space, then We now show that the converse is true whenever - - The relation on X is X x X .
B is closed, so S” - B is open and hence locally pathwise connected. This implies that the path components are components. Let C, and C, be the components of S” - B. Since C, u B is closed, the boundary of C, is contained in B. (Here we mean by the boundary of C , , the set dC, = - C,”). The proof will be complete when we show that B c d C , . Let x E B and U be a neighborhood of x in S”. 13). 28 H*(Sn - ( B - K ) ) = 2 with generator in dimension zero. Thus, S” - ( B - K ) has one path component.