The fundamental group of a $p$-compact group

W. G. Dwyer and C. W. Wilkerson

dwyer.1@nd.edu
wilker@math.purdue.edu

The notion of a $p$-compact group is a homotopy theoretic version of the
geometric or analytic notion of compact Lie group, although the homotopy
theory differs from the geometry is that there are parallel theories of
$p$-compact groups, one for each prime number~$p$. A key feature of the
theory of compact Lie groups is the relationship between centers and
fundamental groups; these play off against one another, at least in the
semisimple case, in that the center of the simply connected form is the
fundamental group of the adjoint form. There are explicit ways to
compute the center or fundamental group of a compact Lie group in terms
of the normalizer of the maximal torus.  For some time there has in fact
been a corresponding formula for the center of $p$-compact groups, but
in general the fundamental group has eluded analysis.  The purpose of
the present paper is to remedy this deficit.

For any space $Y$, we let $\HZp_i(Y)$ denotes $\lim{}_n\HH_i(Y;\Z/p^n)$.
Suppose that $X$ is a connected $p$-compact group, with maximal torus
$T$ and torus normalizer $\NT$. It is known that the map 
$\pi_1(T)\to\pi_1(X)$ is surjective or equivalently
that the map $\HZp_2(\BB T)\to\HZp_2(\BB X)$ is surjective.
We prove the following statement.

Main Theorem: If $X$ is a connected $p$-compact group, then the kernel
  of the map $\HZp_2 \BB T\to \HZp_2\BB\NT$ is the same as the
  kernel of the map $\HZp_2 \BB T\to \HZp_2\BB X$. Equivalently, the
  image of the map $\HZp_2\BB T\to \HZp_2(\BB\NT)$ is (naturally)
  isomorphic to $\pi_1X$.

  There is a proof of the corresponding statement for compact Lie groups
  which relies on the Feshbach double coset formula Our proof of the
  MainTheorem uses a transfer calculation that in practice amounts to a
  weak homological reflection of the double coset formula; we can get
  away with this because we have a splitting of $\HZp_2(\BB\NT)$.

  It is possible to derive from the MainTheorem a more explicit formula
  for $\pi_1X$; this formula is known for $p$~odd as a consequence of
  the classification theorem for $p$ odd. Our demonstration does not use
  the classification theorem.

  Let $W$ denote the Weyl group of $X$. If $p$ is odd, then $\pi_1X$ is
  naturally isomorphic to the module of coinvariants $\HH_0(W;\HZp_2(\BB
  T))$ .  If $p=2$, then up to factors which do not contribute to
  $\pi_1X$, the normalizer of the torus in $X$ is derived by
  $\Ftwo$-completion from the normalizer $\NT_G$ of a maximal torus
  $T_G$ in a connected compact Lie group~$G$ . The image of the map
  $\HH_2(\BB T_G;\Z)\to\HH_2(\BB\NT_G;\Z)$ is isomorphic to $\pi_1G$ ,
  and so by the MainTheorem the tensor product of this image with
  $\Ztwo$ is $\pi_1X$. This image can be computed from the marked
  reflection lattice $(\pi_1T_G, \{b_\sigma,\beta_\sigma\})$
  corresponding to the root system of $G$ or, after tensoring with
  $\Ztwo$, from the marked complete reflection lattice
  $(\pi_1T,\{b_\sigma,\beta_\sigma\})$ associated to $X$ The upshot is
  that $\pi_1X$ is the quotient of $\pi_1T=\pi_2\BB T=\HZtwo_2\BB T$ by
  the $\Ztwo$--submodule generated by the elements $\{b_\sigma\}$.

  Another way to describe this calculation is the following. For each
  reflection $s_\alpha$ in the Weyl group~$W$, let $u_\alpha$ be a
  generator over $\Zp$ of the rank~1 submodule of $\pi_1T$ given by the
  image of $(1-s_\alpha)$. If $p$ is odd let $v_\alpha=u_\alpha$; if
  $p=2$, let $v_\alpha=u_\alpha$ or $u_\alpha/2$, according to whether
  the marking of $s_\alpha$ is trivial or non-trivial. Then $\pi_1X$ is
  the quotient of $\pi_1T$ by the $\Zp$-span of the
  elements~$v_\alpha$. See the upcoming even classification by Andersen
  and Grodal for more details.