Hodge and signature theorems for a family of manifolds with fibration boundary

Abstract

Let $\bar{M}$ be a manifold with boundary $Y$ which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, $x$. We prove $L^2$ Hodge and signature theorems for $M$ endowed with a metric of the form $dx^2 + x^{2c} h + k$, where $k$ is the lift to $Y$ of the metric on the base of the fibre bundle, $h$ is a two form on $Y$ which restricts to a metric on each fibre, and $0 \leq c \leq 1$. These metrics interpolate between the case when $c=0$, in which case the metric near the boundary is a cylinder, and the case where $c=1$, in which case the metric near the boundary is that of a cone bundle over the base of the boundary fibration. We show that the $L^2$ Hodge theorems for the cohomologies given by the maximal and minimal extensions of $d$ with respect to these metrics and the $L^2$ signature theorem for the image of the minimal cohomology in the maximal cohomology interpolate between known results for $L^2$ Hodge and signature theorems for cylindrical and cone bundle type metrics. In particular, the Hodge theorems all relate the related spaces of $L^2$ harmonic forms to intersection cohomology of varying perversities for $X$, the space formed from $\bar{M}$ by collapsing the fibres of $Y$ at the boundary. The signature theorem involves variations on the $τ$ invariant described by Dai.