Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum $s$-$t$ cut, the minimum multiway cut, and the minimum $k$-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial $r$-cycle $ζ$ in a simplicial complex $\mathsf{K}$, find a set $\mathcal{S}$ of $r$-dimensional simplices of minimum cardinality so that $\mathcal{S}$ meets every cycle homologous to $ζ$. Our main result is that this problem admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size $k$. On the positive side, we show that THS admits an FPT algorithm with respect to $k+d$, where $d$ is the maximum degree of the Hasse graph of the complex $\mathsf{K}$. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding $r$-cycle $ζ$ in a simplicial complex $\mathsf{K}$, find a set $\mathcal{S}$ of $(r+1)$-dimensional simplices of minimum cardinality so that the removal of $\mathcal{S}$ from $\mathsf{K}$ makes $ζ$ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an $O(\log n)$-approximation FPT algorithm for $(r+1)$-dimensional complexes with the $(r+1)$-th Betti number $β_{r+1}$ as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.