  The null space of the $k$-th order Laplacian $\mathbf{\mathcal L}_k$, known
as the {\em $k$-th homology vector space}, encodes the non-trivial topology of
a manifold or a network. Understanding the structure of the homology embedding
can thus disclose geometric or topological information from the data. The study
of the null space embedding of the graph Laplacian $\mathbf{\mathcal L}_0$ has
spurred new research and applications, such as spectral clustering algorithms
with theoretical guarantees and estimators of the Stochastic Block Model. In
this work, we investigate the geometry of the $k$-th homology embedding and
focus on cases reminiscent of spectral clustering. Namely, we analyze the {\em
connected sum} of manifolds as a perturbation to the direct sum of their
homology embeddings. We propose an algorithm to factorize the homology
embedding into subspaces corresponding to a manifold's simplest topological
components. The proposed framework is applied to the {\em shortest homologous
loop detection} problem, a problem known to be NP-hard in general. Our spectral
loop detection algorithm scales better than existing methods and is effective
on diverse data such as point clouds and images.
