  Topological states of matter are, generally, quantum liquids of conserved
topological defects. We establish this by constructing and analyzing
topological field theories which describe the dynamics of field singularities
using gauge fields. Homotopy groups are utilized to identify topologically
protected singularities, and the conservation of their protected number is
captured by a topological action term that unambiguously obtains from the given
set of symmetries. Stable phases of these theories include quantum liquids with
emergent massless Abelian and non-Abelian gauge fields, as well as topological
orders with long-range quantum entanglement, fractional excitations, boundary
modes and unconventional responses to external perturbations. This paper
focuses on the derivation of topological field theories and basic
phenomenological characterization of topological orders associated with
homotopy groups $\pi_n(S^n)$, $n\ge 1$. These homotopies govern monopole and
hedgehog topological defects in $d=n+1$ dimensions, and enable the
generalization of both weakly-interacting and fractional quantum Hall liquids
of vortices to $d>2$. Hedgehogs have not been in the spotlight so far, but they
are particularly important defects of magnetic moments because they can be
stimulated in realistic systems with spin-orbit coupling, such as chiral
magnets and $d=3$ topological materials. We predict novel topological orders in
systems with U(1)$\times$Spin($d$) symmetry in which fractional electric charge
attaches to hedgehogs. Monopoles, the analogous defects of charge or generic
U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge
in purely magnetic systems. The latter can lead to spin liquids with different
topological orders than that of the RVB spin liquid. Charge fractionalization
of quarks in atomic nuclei is also seen as possibly arising from the
charge-hedgehog attachment.
