A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on Z/2[3]--framed immersions, and quaternionic-cyclic structure on Z/2[4]--framed immersions. Using these concepts, we prove that for sufficiently large n
Nearby in the stack
,
n=2ℓ−2
, an arbitrary skew-framed immersion in Euclidean
n
-space
Rn
has zero Kervaire invariant. Additionally, for
ℓ≥12
(i.e., for
n≥4094
) an arbitrary skew-framed immersion in Euclidean
n
-space
Rn
has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16.