The main result of the article concerns the stable mod \(p\) Hurewicz image of classifying spaces of elementary abelian \(p\)-groups \(V\) of finite rank, say \(s\). More precisely, if \(n\) is a natural number and \(p\) a prime the author gives an algebraic condition on \(s,p\) and \(n\) which forces the stable Hurewicz map \(\pi^s_{2n-s}(BV_+) \to H_{2n-s}(BV;Z/p)\) to be trivial. His result puts a rather strong condition on the stable Hurewicz image, e.g. taken together with the Kahn-Priddy theorem [D. S. Kahn and S. B. Priddy, Bull. Am. Math. Soc. 78, 981-987 (1972; Zbl 0265.55009)] it implies the Adams Hopf invariant one theorem. It also implies that for all \(s\) there is some integer \(n(s)\) such that no non-trivial element in the image of \[ ({\mathcal P}^0)^{n(s)}(\text{Ext}_{A_*}^{s,*}(Z/p,Z/p)) \subset \text{Ext}_{A_*}^{s,p^{n(s)}*}(Z/p,Z/p) \] comes from \(\pi^s_{p^{n(s)*}-s}(BV_+)\) via the \(s\)-fold iterated transfer. The last statement is called the iterated transfer analogue of the new doomsday conjecture.
To obtain his result the author is looking at the map \(BP_{2n-s}BV \to H_{2n-s}(BV;Z/p)\) and the induced map on the primitive parts \(P(BP_{2n-s}BV) \to P(H_{2n-s}(BV;Z/p))\). He generalizes Kamenko’s reduction theorem to arbitrary prime numbers and uses it in conjunction with the Adams spectral sequence to conclude that elements in \(P(BP_{2n-s}BV)\) with non-trivial image in \(P(H_{2n-s}(BV;Z/p))\) must have high \(p\)-exponent provided the algebraic condition on \(s,p\) and \(n\) holds. On the other hand he shows (using \(BP\)-Adams operations) that the algebraic condition guarantees that such elements must have comparatively small \(p\)-exponent. It follows that the map \(P(BP_{2n-s}BV) \to P(H_{2n-s}(BV;Z/p))\) must be the trivial map, which then implies the result.