  This article investigates the parameter space of the exponential family
$z\mapsto \exp(z)+\kappa$. We prove that the boundary (in $\C$) of every
hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as
well as Baker and Rippon. In fact, we prove the stronger statement that the
exponential bifurcation locus is connected in $\C$, which is an analog of
Douady and Hubbard's celebrated theorem that the Mandelbrot set is connected.
We show furthermore that $\infty$ is not accessible through any nonhyperbolic
("queer") stable component.
  The main part of the argument consists of demonstrating a general "Squeezing
Lemma", which controls the structure of parameter space near infinity. We also
prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees
of hyperbolic components.
