  In this paper we consider the semilinear Cauchy problem for the heat equation
with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat
operator is given in this case by $\partial_t-\Delta_H$, where $\Delta_H$ is
the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita
exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension
of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in
time solutions in the subcritical case and in the critical case. In order to
get the upper bound estimate for the lifespan (especially, in the critical
case) we employ a revisited test function method developed recently by
Ikeda-Sobajima. On the other hand, to find the lower bound estimate for the
lifespan we prove a local in time result in weighted $L^\infty$ space.
