Dimensionality reduction is an integral part of data visualization. It is a process that obtains a structure preserving low-dimensional representation of the high-dimensional data. Two common criteria can be used to achieve a dimensionality reduction: distance preservation and topology preservation. Inspired by recent work in topological data analysis, we are on the quest for a dimensionality reduction technique that achieves the criterion of homology preservation, a generalized version of topology preservation. Specifically, we are interested in using topology-inspired manifold landmarking and manifold tearing to aid such a process and evaluate their effectiveness.