1. The problem asks whether the parabola $y = x^2$ and the $x$-axis are topologically equivalent to a cup and a donut, and if there are simplicial or sheaf methods in computational topology to decompose and reconstruct the parabola from the $x$-axis. 2. First, let's clarify the topology: a donut (torus) is a surface with a hole, while a cup is topologically a disk with a handle. The $x$-axis is homeomorphic to the real line $\mathbb{R}$, and the parabola $y = x^2$ is also homeomorphic to $\mathbb{R}$ because it is a continuous curve without holes. 3. Therefore, topologically, the $x$-axis and the parabola $y = x^2$ are equivalent (both are homeomorphic to $\mathbb{R}$), but neither is equivalent to a donut or a cup, which have fundamentally different topological properties (holes and genus). 4. Regarding computational topology, simplicial complexes and sheaf theory are powerful tools to study topological spaces. A parabola can be discretized into a simplicial complex (a graph or a set of vertices and edges) that approximates its shape. 5. Decomposition methods in computational topology often involve breaking a space into simpler pieces (like simplices) and studying their connectivity. Sheaf theory can encode local data and how it glues globally, useful for reconstructing spaces from local information. 6. In practice, one can represent the parabola as a simplicial complex and project it onto the $x$-axis (a simpler 1D complex). The inverse procedure reconstructs the parabola by lifting points from the $x$-axis using the function $f(x) = x^2$. 7. This process is not a topological equivalence to a donut or cup but a computational method to analyze and reconstruct curves using algebraic topology tools. Final answer: The parabola and the $x$-axis are topologically equivalent to each other (both homeomorphic to $\mathbb{R}$), but not to a cup or donut. Simplicial and sheaf methods can be used to decompose and reconstruct the parabola from the $x$-axis in computational topology, but this is a functional and combinatorial procedure, not a topological equivalence to surfaces like a donut.