  We study topology of frustration in $d$-dimensional Ising spin glasses with
$d\ge 2$ with nearest-neighbor interactions. We prove the following: For any
given spin configuration, the domain walls on the unfrustration network are all
transverse to a frustrated loop on the unfrustration network, where a domain
wall is defined to be a connected element of the collection of all the
$(d-1)$-cells which are dual to the bonds having an unfavorable energy, and the
unfrustration network is the collection of all the unfrustrated plaquettes.
These domain walls are topologically nontrivial because they are all related to
the global frustration of a loop on the unfrustration network. Taking account
of the thermal stability for the domain walls, we can explain the numerical
results that three or higher dimensional systems exhibit a spin glass phase,
whereas two-dimensional ones do not. Namely, in two dimensions, the thermal
fluctuations of the topologically nontrivial domain walls destroy the order of
the frozen spins on the unfrustration network, whereas they do not in three or
higher dimensions. This may be interpreted as a global, topological effect of
the frustrations.
