| Universal Coefficient Theorem |
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| CATEGORIES ABOUT UNIVERSAL COEFFICIENT THEOREM | |
| algebraic topology | |
| homological algebra | |
| mathematical theorems | |
H do in a certain, definite sense determine the groups H
For example it is common to take ''A'' to be ''Z/2Z'', so that coefficients are mod 2. This becomes straightforward in the absence of 2- Torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti Number s ''b''''i'' of ''X'' and the Betti numbers ''b''''i'',''F'' with coefficients in a Field ''F''. These can differ, but only when the Characteristic of ''F'' is a Prime Number ''p'' for which there is some ''p''-torsion in the homology. The statement of the universal coefficient theorem runs as follows: consider : where ''H''''i'' means ''H''''i''(''X'',''Z''). Then there is an Injective Group Homomorphism ι from it to ''H''''i''(''X'',''A''). The theorem describes the Cokernel of ι as Tor This ''Tor'' group can therefore be described as the obstruction to ι being an isomorphism, which could be thought of as the 'expected' result. There is also a universal coefficient theorem for cohomology, involving the Ext Functor . |
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