  It has been proposed that cobordism and K-theory groups, which can be
mathematically related in certain cases, are physically associated to
generalised higher-form symmetries. As a consequence, they should be broken or
gauged in any consistent theory of quantum gravity, in accordance with
swampland conjectures. We provide further support to this idea by showing that
cobordism and K-theory groups of a general manifold $X$ reproduce the pattern
of symmetries expected from the dimensional reduction of the theory on $X$, as
well as their breaking and gauging. To this end, we employ the
Atiyah-Hirzebruch spectral sequence to compute such groups for common choices
of $X$ in string compactifications.
