Cohomology ring of BG - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:02:24Z http://mathoverflow.net/feeds/question/116894 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116894/cohomology-ring-of-bg Cohomology ring of BG Paul Siegel 2012-12-20T19:10:01Z 2012-12-20T20:30:11Z <p>Let $G$ be a compact Lie group, let $T$ be a maximal torus, and let $W$ be the Weyl group. My main question is as follows: </p> <ul> <li>How does one prove that <code>$H^\ast(BG,\mathbb{Q})$</code> is isomorphic to the $W$-invariant part of $H^\ast(BT,\mathbb{Q}) \cong \mathbb{Q}[[x_1, \ldots, x_n]]$? This is apparently basic knowledge in algebraic topology, because I keep reading "recall that..." followed by some version of this statement and no references. But I can't find a proof in any of my textbooks.</li> </ul> <p>I would ideally like a reference which also addresses the following secondary question:</p> <ul> <li>When is the natural map $H^\ast(BG,\mathbb{Z}) \to H^\ast(BT,\mathbb{Z})^W$ an isomorphism, and what can one say about the integral cohomology ring of $BG$ when it is not? Note the fact that the map above is an isomorphism for $G = U(n)$ is equivalent to the statement that the Chern classes are integral.</li> </ul> <p>Thanks!</p> http://mathoverflow.net/questions/116894/cohomology-ring-of-bg/116903#116903 Answer by Craig Westerland for Cohomology ring of BG Craig Westerland 2012-12-20T20:03:18Z 2012-12-20T20:03:18Z <p>Notice that there is a sequence of homomorphisms $T \to N \to G$, where $N$ is the maximal torus normaliser (so $W = N/T$). $W$ acts on $BT$ (because it acts on $T$ by conjugation through group homomorphisms), and there is an equivalence from the classifying space of $N$ to the Borel construction for this action:</p> <p>$$BN \simeq EW \times_W BT.$$</p> <p>Consequently, we can compute the cohomology of $BN$ from the Leray-Serre spectral sequence</p> <p>$$H^\ast(W; H^\ast(BT)) \implies H^\ast(BN).$$</p> <p>Taking rational cohomology, this spectral sequence is concentrated in group-cohomological degree $0$, since $W$ is a finite group. Therefore the spectral sequence collapses at $E_2$, which is $H^0(W, H^\ast BT) = H^\ast(BT)^W$.</p> <p>It therefore suffices to show that the map $BN \to BG$ is an isomorphism in rational cohomology. If we write $BN$ as $EG / N$, this map is a fibre bundle with fibre $G / N$, so it's enough to show that $G/N$ has the rational homology of a point. </p> <p>For instance, if $G = SU(2)$, $N = \mathbb{Z} / 2 \ltimes T$, and $T = S^1$. Then $G/T = \mathbb{C} P^1$, and the action of $\mathbb{Z} / 2$ is antipodal, giving $G / N = \mathbb{R} P^2$, which is indeed rationally a point. I don't remember the argument in general, but I think this is always true.</p> <p>Hopefully this indicates how the corresponding integral statement can fail - there can be torsion contributions from the higher group cohomology of $W$, which needs to be exactly cancelled (via a differential in the second spectral sequence above) with a torsion cohomology class from $G/N$.</p> http://mathoverflow.net/questions/116894/cohomology-ring-of-bg/116904#116904 Answer by Chris Gerig for Cohomology ring of BG Chris Gerig 2012-12-20T20:15:07Z 2012-12-20T20:15:07Z <p>I don't remember where I heard the following proof/sketch:</p> <p>Using the fibering $G/T\to BT\to BG$ and the fact that the Euler class of $G/T$ is nonzero, we have that $H^\ast(BG)$ embeds into $H^\ast(BT)$ (it composes with the transfer map to be multiplication by the Euler class); and the desired isomorphism comes from the fact that $W$ acts on $H^*(G/T)$ as the regular representation. </p> <p>This is actually a special case of equivariant cohomology, where we instead use the Borel construction and the fibering $G/T\to M_T\to M_G$.</p> http://mathoverflow.net/questions/116894/cohomology-ring-of-bg/116907#116907 Answer by Ralph for Cohomology ring of BG Ralph 2012-12-20T20:30:11Z 2012-12-20T20:30:11Z <p><strong>Q1:</strong> Let me first note, that the statement $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BT,\mathbb{Q})^W\tag{\ast}$$ made in the question requires in addition that $G$ is connected. However, the general case can be reduced to the connected case since $$H^\ast(BG,\mathbb{Q}) \cong H^\ast(BG_0,\mathbb{Q})^{G/G_0}$$ where $G_0$ is the identity component of $G$. </p> <p>A text book reference for $(\ast)$ can be found in [Hsiang: Cohomology theory of topological transformation groups, Chapter III, §1, Lemma 1.1]. The results of the book that are relevant for your question can also be found in the following paper: <a href="http://www.math.uwo.ca/~rgonzal3/qfy.pdf" rel="nofollow">http://www.math.uwo.ca/~rgonzal3/qfy.pdf</a> (cf. Remark 9, Lemma 5). Other approaches and more information can be found in </p> <p>$\quad$<a href="http://mathoverflow.net/questions/61784/cohomology-of-bg-g-compact-lie-group/61796#61796" rel="nofollow">http://mathoverflow.net/questions/61784/cohomology-of-bg-g-compact-lie-group/61796#61796</a></p> <p><strong>Q2:</strong> First note that the kernel of the restriction map $$\rho^\ast: H^\ast(BG;\mathbb{Z}) \to H^\ast(BT;\mathbb{Z})$$ is the tosion subgroup $Tors$ of $H^\ast(BG;\mathbb{Z})$. So, $\rho^\ast$ is injective iff $H^\ast(BG;\mathbb{Z})$ is torsion-free (there is a lot of literature about torsion of $H^\ast(BG;\mathbb{Z})$ so I won't discuss it here).</p> <p>A short paper of Feshbach [The image of $H^\ast(BG;\mathbb{Z})$ in $H^\ast(BT;\mathbb{Z})$ for $G$ a compact Lie group with maximal torus $T$. Topology 20(1981),93-95] characterizes when the induced map $$\bar{\rho}^\ast: H^\ast(BG;\mathbb{Z})/Tors \to H^\ast(BT;\mathbb{Z})$$ is an isomorphism: </p> <p>$\quad\bar{\rho}^\ast$ is an isomorphism iff $H^\ast(BG;\mathbb{Z})/Tors \otimes \mathbb{F}_p$ is an integral domain for each prime $p$. </p> <p>There is also a counter-example that shows that $\bar{\rho}^\ast$ is not always an isomorphism. </p>