reference for (co)homology theories - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:05:50Z http://mathoverflow.net/feeds/question/114201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114201/reference-for-cohomology-theories reference for (co)homology theories seub 2012-11-23T00:33:04Z 2012-11-25T20:02:00Z <p>Hi everyone,</p> <p>Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it.</p> <p>I was wondering if someone could recommend a cool reference that would, ideally:</p> <ol> <li>Cover several cohomology theories, including, say: singular, de Rham, Dolbeault, group cohomology, sheaf cohomology.</li> <li>Explain the relations between these theories.</li> <li>Not too hardcore to read (not too formal, containing examples, etc.)</li> </ol> <p>Thanks for helping!</p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114208#114208 Answer by paul garrett for reference for (co)homology theories paul garrett 2012-11-23T04:49:36Z 2012-11-23T04:49:36Z <p>C. Weibel's Intro to hom alg is likely what one wants for a perspective on how these different examples fit into a larger context. In particular, thinking that the relevant algebra is dismissible is misguided: many of the seeming geometric issues are more shallow, in a good sense.</p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114221#114221 Answer by Thomas Rot for reference for (co)homology theories Thomas Rot 2012-11-23T08:20:18Z 2012-11-23T09:32:40Z <p>I really like the book of Bott and Tu for the De Rham theory. Hatchers book - freely available on his site - contains nice treatments of singular and cellular (co) homologies.</p> <p>About your comment. What relations between the theories are you looking for? The Eilenberg Steenrod axioms - <a href="http://www.encyclopediaofmath.org/index.php/Steenrod-Eilenberg_axioms" rel="nofollow">http://www.encyclopediaofmath.org/index.php/Steenrod-Eilenberg_axioms</a> - show that the singular, cellular, and de Rham theories are the same (you have to be a bit careful with the coefficients of course), on spaces where they are all defined. I believe the book of Bredon discusses this a bit, but I don't have it with me here (there is a very short passage in Hatcher). Group cohomology can be seen to be the cohomology of a certain space associated to the group. I does not really matter in which theory we compute these, because they will give the same results. I'm not familiar with the Daubault and sheaf cohomology, so I don't have anything relevant to add to this.</p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114225#114225 Answer by daniele for reference for (co)homology theories daniele 2012-11-23T09:26:30Z 2012-11-23T09:26:30Z <p>what about this: <em>Une introduction aux (co)homologies, Cours &amp; exercices</em> by T. Masson: <a href="http://science.thilucmic.fr/spip.php?article16" rel="nofollow">http://science.thilucmic.fr/spip.php?article16</a></p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114257#114257 Answer by Donu Arapura for reference for (co)homology theories Donu Arapura 2012-11-23T15:57:03Z 2012-11-24T17:34:50Z <p>Let me make a few comments here starting with sheaf cohomology. It can be defined in several ways, but the definitions are less important the properties:</p> <ol> <li>$H^0$ of a sheaf coincides with its global sections.</li> <li>Short exact sequences of sheaves give rise to long exact sequences of sheaf cohomology.</li> <li>It can be computed by acyclic resolutions, such as fine resolutions on a paracompact Hausdorff space.</li> </ol> <p>Using 3 plus the Poincaré lemma, we get a version of de Rham's theorem that sheaf cohomology of a manifold with coefficients in the constant sheaf $\mathbb{R}$ coincides with de Rham cohomology. A similar application of 3 shows that on a complex manifold, sheaf cohomology of the sheaf of holomorphic $p$-forms coincides with Dolbeault cohomology. Also, on a sufficiently nice space (e.g. a manifold) singular cohomology with coefficients in an abelian group $A$ coincides with sheaf cohomology with coefficients in $A$. This is also true if $A$ is replaced by a local system. A local system comes with a monodromy representation $\pi_1(X)\to Aut(A)$. There is map from group cohomology to singular cohomology $$H^i(\pi_1(X), A) \to H^i(X, A)$$ which is an isomorphism if $X$ is aspherical (Eilenberg-Maclane) but not in general (e.g. take $X=S^n, n>1$).</p> <p><strong>Postscript</strong> A more genuine answer, which would be even more unhelpful to the OP than the one I gave, is that I doubt there is a single coherent and easy treatment of all of these disparate topics. But there are plenty of good references (some of which have been mentioned) that cover algebraic topology, group cohomology and its applications, sheaf cohomology and its applications, homological algebra...</p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114373#114373 Answer by René Pannekoek for reference for (co)homology theories René Pannekoek 2012-11-24T20:23:01Z 2012-11-24T20:23:01Z <p>I learned sheaf cohomology from Claire Voisin's <em>Hodge Theory and Complex Algebraic Geometry I</em>. This is a great book. As its name suggests, it also spends quite some time explaining Dolbeault cohomology, De Rham cohomology, singular cohomology, and how all these are defined/can be understood in terms of sheaf cohomology. (There's no group cohomology, as far as I recall.)</p> http://mathoverflow.net/questions/114201/reference-for-cohomology-theories/114443#114443 Answer by Deane Yang for reference for (co)homology theories Deane Yang 2012-11-25T20:02:00Z 2012-11-25T20:02:00Z <p>Chapter 5 of Frank Warner's book, Foundations of Differentiable Manifolds, (http://www.amazon.com/Foundations-Differentiable-Manifolds-Graduate-Mathematics/dp/0387908943) presents four different cohomology theories and why they are all isomorphic.</p>