  In this paper a toy model of quantum topology is reviewed to study effects of
matter and gauge fields on the topology fluctuations. In the model a collection
of N one dimensional manifolds are considered where a set of boundary
conditions on states of Hilbert space specifies a set of all topologies
perceived by quantum particle and probability of having a specific topology is
determined by a partition function over all the topologies in the context of
noncommutative spectral geometry. In general the topologies will be fuzzy with
the exception of a particular case which is localized by imposing a specific
boundary condition. Here fermions and bosons are added to the model. It is
shown that in the presence of matter, the fuzziness of topology will be
dependent on N, however for large N the dependence is removed similar to the
case without matter. Also turning on a particular background gauge field, can
overcome the fuzziness of topology to reach a localized topology with classical
interpretation. It can be seen that for large N more opportunities can be
provided for choosing the background gauge field to localize the fuzzy
topology.
