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Let $\bar{M}$ be a manifold with boundary $Y$ which is the total space of afibre 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 metricof the form $dx^2 + x^{2c} h + k$, where $k$ is the lift to $Y$ of the metricon the base of the fibre bundle, $h$ is a two form on $Y$ which restricts to ametric on each fibre, and $0 \leq c \leq 1$. These metrics interpolate betweenthe 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 ofa 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 extensionsof $d$ with respect to these metrics and the $L^2$ signature theorem for theimage of the minimal cohomology in the maximal cohomology interpolate betweenknown results for $L^2$ Hodge and signature theorems for cylindrical and conebundle type metrics. In particular, the Hodge theorems all relate the relatedspaces of $L^2$ harmonic forms to intersection cohomology of varyingperversities for $X$, the space formed from $\bar{M}$ by collapsing the fibresof $Y$ at the boundary. The signature theorem involves variations on the $\tau$invariant described by Dai.Comment: 24 pages, 3 diagrams missing from end of document due to technical  difficultie

Hodge and signature theorems for a family of manifolds with fibre bundle boundary