Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. Products in stable homotopy correspond to cartesian products in framed bordism, so $\eta^k$ is represented by the torus $T^k$ with its Lie group framing. In this question Chris Schommer-Pries gives a description of a specific nullbordism of $T^4$ representing the equation $\eta^4=0$ using the $K3$ surface.
My question: Is there a nullbordism of $T^4$ (or more generally any $T^k$ for $k\geq 4$) with an action of $T^4$ (of $T^k$) that when restricted to the boundary gives the usual left-translation action of $T^4$ (of $T^k$) on itself?
