词条 | Universal coefficient theorem |
释义 |
In algebraic topology, universal coefficient theorems establish relationships between homology and cohomology theories. For instance, the integral homology theory of a topological space {{mvar|X}}, and its homology with coefficients in any abelian group {{mvar|A}} are related as follows: the integral homology groups {{math|Hi(X; Z)}} completely determine the groups {{math|Hi(X; A)}} Here {{math|Hi}} might be the simplicial homology or more general singular homology theory: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients {{mvar|A}} may be used, at the cost of using a Tor functor. For example it is common to take {{mvar|A}} to be {{math|Z/2Z}}, so that coefficients are modulo 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 numbers {{math|bi}} of {{mvar|X}} and the Betti numbers {{math|bi,F}} with coefficients in a field {{mvar|F}}. These can differ, but only when the characteristic of {{mvar|F}} is a prime number {{mvar|p}} for which there is some {{mvar|p}}-torsion in the homology. Statement of the homology caseConsider the tensor product of modules {{math|Hi(X; Z) ⊗ A}}. The theorem states there is a short exact sequence Furthermore, this sequence splits, though not naturally. Here {{mvar|μ}} is a map induced by the bilinear map {{math|Hi(X; Z) × A → Hi(X; A)}}. If the coefficient ring {{mvar|A}} is {{math|Z/pZ}}, this is a special case of the Bockstein spectral sequence. Universal coefficient theorem for cohomologyLet {{mvar|G}} be a module over a principal ideal domain {{mvar|R}} (e.g., {{math|Z}} or a field.) There is also a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence As in the homology case, the sequence splits, though not naturally. In fact, suppose and define: Then {{mvar|h}} above is the canonical map: An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map {{mvar|h}} takes a homotopy class of maps from {{mvar|X}} to {{math|K(G, i)}} to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.[1] Example: mod 2 cohomology of the real projective spaceLet {{math|X {{=}} Pn(R)}}, the real projective space. We compute the singular cohomology of {{mvar|X}} with coefficients in {{math|R {{=}} Z/2Z}}. Knowing that the integer homology is given by: We have {{math|Ext(R, R) {{=}} R, Ext(Z, R) {{=}} 0}}, so that the above exact sequences yield In fact the total cohomology ring structure is CorollariesA special case of the theorem is computing integral cohomology. For a finite CW complex {{mvar|X}}, {{math|Hi(X; Z)}} is finitely generated, and so we have the following decomposition. where {{math|βi(X)}} are the Betti numbers of {{mvar|X}} and is the torsion part of . One may check that and This gives the following statement for integral cohomology: For {{mvar|X}} an orientable, closed, and connected {{mvar|n}}-manifold, this corollary coupled with Poincaré duality gives that {{math|βi(X) {{=}} βn−i(X)}}. Notes1. ^{{Harv|Kainen|1971}} References
| last = Kainen | first = P. C. | authorlink = Paul Chester Kainen | coauthors = | title = Weak Adjoint Functors | journal = Mathematische Zeitschrift | volume = 122 | issue = | pages = 1–9 | publisher = | year = 1971 | pmid = | pmc = | doi = 10.1007/bf01113560 External links
2 : Homological algebra|Theorems in algebraic topology |
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