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Unformatted text preview: Math 872 Algebraic Topology Running lecture notes Homology theory: Fundamental groups are a remarkably powerful tool for studying spaces; they capture a great deal of the global structure of a space, and so they are very good a distinguishing between homotopyinequivalent spaces. In theory! But in practice, they suffer from the fact that deciding whether two groups are isomorphic or not is, in general, undecideable! Homology theory is designed to get around this deficiency; the theory, by design, builds (a sequence of) abelian groups H i ( X ) from a topological space. And deciding whether or not two abelian groups, at least if you’re given a presentation for them, is, in the end, a matter of fairly routine linear algebra. Mostly because of the Fundamental Theorem of Finitelygenerated Abelian groups; each such has a unique representation as Z m ⊕ Z m 1 ⊕ ··· ⊕ Z m n with m i +1  m i for every i . There are also “higher” homotopy groups beyond the fundamental group π 1 , (hence the name pi one ); elements are homtopy classes, rel boundary, of based maps ( I n , ∂I n ) → ( X, x ). Multiplication is again by concatenation. But unlike π 1 , where we have a chance to compute it via Seifertvan Kampen, nobody, for example knows what all of the homotopy groups π n ( S 2 are (except that nearly all of them are nontrivial!). Like π 1 , it describes, essentially, maps of S n into X which don’t extend to maps of D n +1 , i.e., it turns the “ ndimensional holes” of X into a group. Homology theory does the exact same thing, counting ndimensional holes. In the end we will find it to be extremely computable; but it will require building a fair bit of machinery before it will become so transparent to calculate. But the short version is that the homology groups compute “cycles mod boundaries”, that is, ndimesional objects/subsets that have no boundary (in the appropriate sense) modulo objects that are the boundary of ( n + 1) dimensional ones. There are, in fact, probably as many ways to define homology groups as there are people actively working in the field; we will focus on two, simplicial homology and singular homology. The first is quick to define and compute, but hard to show is an invariant! The second is quick to see is an invariant, but, on the face of it, hard to compute! Luckily, for spaces where they are both defined, they are isomorphic. So, in the end, we get an invariant that is quick to compute. Of course, so is the invariant “4”; but this one will be a bit more informative.... First, simplicial homology. This is a sequence of groups defined for spaces for which they are easiest to define, which Hatcher calls Δcomplexes. Basically, they are spaces defined by gluing simplices together using nice enough maps. More precisely, the standard nsimplex Δ n is the set of points { ( x 1 , . . .x n +1 ) ∈ R n +1 : ∑ x i = 1 , x i ≥ 0 for all i } . This can also be expressed as convex linear combinations (literally, that’s the conditions on the...
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.
 Spring '11
 Aster
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