By Andrew H. Wallace
ISBN-10: 0486457869
ISBN-13: 9780486457864
This self-contained therapy assumes just some wisdom of genuine numbers and actual research. the 1st 3 chapters specialise in the fundamentals of point-set topology, and then the textual content proceeds to homology teams and non-stop mapping, barycentric subdivision, and simplicial complexes. routines shape an essential component of the textual content. 1961 variation.
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Sample text
J 5 / + 1 ) ) . n tf(v) n fr(i) n . . n 3V(a) Pi qr+i, and hence these intersections are correctly connected. n fr(fl) n tfr+i) - Int (reg. n qi{a) n ? r + i ) , hits ^(i) fL-H <7;(a) (1 # r +i in 7i{Sl), eral position argument) A is (/ — l)-connected. # ! n ^ (fl) n(uf = i aox). ,p. fl ^'(o))> and hence £ ; is a (m — a)-ball. n ^( a ) n qr+\ PI 30,- which is a (m — a — l)-ball (a regular neighborhood of a cell in the boundary of a manifold N is a cell whose boundary hits dN in a cell [Hd]).
2. R e m a r k . If Po is a handlebody decomposition of a manifold, then any two associated maps are nomotopic [Waj]. If P 2 is a ^-partition, then any two associated maps are /i-nomotopic. 3. Setting. ), Q0 is a partition on M0 such that all elements of Q0 underlie a subcomplex of Lj, Mf is a (k — l)-connected manifold, Q is a //-partition on M', T 0 : QQ —• Q is an injective function that preserves intersections. 4. Step 1. ] ^0 > Q0 _ > © (/(? o n M[i] = * is a sufficiently large index).
6. hold. The next proposition follows immediately from the definition and some elementary properties of LC _1 -spaces (in (iii) use the lifting technique). 8. Proposition, (i) If Y is a LC^ - 1 -compactum, then there exists e > 0 such that any two e-close maps f,g : X —• Y are /U-nomotopic, (ii) If f,g,h : X —• Y are maps into an LC^ - 1 -compactum 7 : Y —• Yt is a UVk~l-surjection Y, and if f ^ (iii) If between LC^ - 1 -compacta , and f,g : X —-• Y maps from a compactum X with 7 / . ^ 7*7, then / ^ Theorem).
An Introduction to Algebraic Topology by Andrew H. Wallace
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