Abstract. A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian p-groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture. IntroductionDuring the last decade, significant progress has been made in the study of the Together with the Kahn-Priddy theorem [KP], this result implies the Adams Hopf invariant one theorem [Ad1] as a special case (when p = 2 and s = 1). Furthermore, when p = s = 2, this result recovers our previous result [M1], which showed the higher Kervaire invariant one elements θ j (j ≥ 5) are not realized by framed hypersurfaces.Our result has another implication (which was in fact our motivation). For this purpose, we recall the new doomsday conjecture proposed in [M3]:

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We show the "non-existence" results are essential for all the previous known applications of the Bauer-Furuta stable homotopy Seiberg-Witten invariants. As an example, we present a unified proof of the adjunction inequalities.We also show that the nilpotency phenomenon explains why the Bauer-Furuta stable homotopy Seiberg-Witten invariants are not enough to prove 11/8-conjecture.

We prove a nilpotency theorem for the Bauer-Furuta stable homotopy Seiberg-Witten invariants for smooth closed 4-manifolds with trivial first Betti number.

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