Determination of bounds on the dimension of manifolds with involutions fixing $F^n\cup F^4$

Let $M^m$ be an $m$-dimensional, smooth and closed manifold, equipped with a smooth involution $T\text{\hspace{0.17em}⁣}:M^m\to M^m$ fixing submanifolds $F^n$ and $F^4$ of dimensions $n$ and $4$, respectively, where $4<n<m$ and $F^n\cup F^4$ does not bound. We determine the upper bound for $m$, for each $n$. The existence of these bounds is ensured by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, $m⩽\frac{5}{2}n$.