The cohomology groups of $\Omega U(n)$

Question

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?

Answer 1

Given a one-dimensional subspace $L<\mathbb{C}^n$ and a number $z\in S^1\subset\mathbb{C}$ we can define $\rho(L)(z)\colon\mathbb{C}^n\to\mathbb{C}^n$ by $\rho(L)(z)=z$ on $L$ and $\rho(L)(z)=1$ on $L^\perp$. This defines $\rho(L)\colon S^1\to U(n)$ with $\rho(L)(1)=1$, or in other words $\rho(L)\in\Omega U(n)$. We thus have $\rho\colon\mathbb{C}P^{n-1}\to\Omega U(n)$, giving $$ \rho_*\colon H_*(\mathbb{C}P^{n-1}) = \mathbb{Z}\{b_0,\dotsc,b_{n-1}\} \to H_*(\Omega U(n)) $$ On the other hand, the space $\Omega U(n)=\Omega^2BU(n)$ is a double loop space and so is a homotopy commutative $H$-space, so $H_*(\Omega U(n))$ is a commutative ring, and it is not hard to see that the element $b_0\in H_0(\Omega U(n))$ is invertible. Thus, $\rho_*$ extends to give a ring map $$ \phi\colon\mathbb{Z}[b_0,\dotsc,b_{n-1}][b_0^{-1}] \to H_*(\Omega U(n)). $$ It is a standard theorem that this map $\phi$ is an isomorphism. This can be proved using the Serre spectral sequence for the fibration $\Omega U(n)\to PU(n)\to U(n)$, for example. (Here $PU(n)$ is the space of based paths, which is contractible.) It is technically more convenient to study $SU(n)$ rather than $U(n)$, as $\Omega SU(n)$ is connected and of finite type, whereas $\Omega U(n)\simeq \mathbb{Z}\times\Omega SU(n)$. As the homology of $\Omega U(n)$ is free, the cohomology is just the dual and is torsion free. However, $H^*(\Omega U(n))$ is not actually free; even in dimension $0$ we have $\prod_{n\in\mathbb{Z}}\mathbb{Z}$, which is not a free abelian group.

Answer 2

Here are some more possible approaches to show that the cohomology of $\Omega U(n)$ is torsion-free, as a complement to Neil Strickland's answer:

In general, a useful tool to compute the cohomology of loop spaces is the Eilenberg-Moore spectral sequence, of which you can find an overview in McCleary's book "A user's guide to spectral sequences". It may well be (I do not have my copy handy) that the particular example of $\Omega U(n)$ is discussed in there.

In the specific case of loop spaces of compact Lie groups, there is a cell decomposition of $\Omega G$ due to Bott, cf. R. Bott: An application of the Morse-theory to the topology of Lie groups", BSMF 84, 251-281. Another approach for the computation of this cell structure using the Bruhat decomposition, cf. H. Garland and M.S. Raghunathan: A Bruhat decomposition for the loop space of a compact group: a new approach to a result of Bott. There is a lot more literature on the homotopy type of $\Omega U(n)$ refining cohomology computations. For instance, there is a stable splitting of $\Omega U(n)$ (which refines and is built on Bott's cell structure mentioned above), cf. M. Crabb: On the stable splitting of $U(n)$ and $\Omega U(n)$. LNMA 1298, 35-53.

There is also a Langlands type approach to determine the homology of loop groups, cf. Z. Yun and X. Zhu: Integral homology of loop groups via Langlands dual groups; DOI: 10.1090/S1088-4165-2011-00399-X, arXiv: 0909.5487.