Abstract. Let T r k be the algebraic transfer that maps from the coinvariants of certain GL k -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV k ) + ) → π S * (S 0 ). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that T r k is an isomorphism for k = 1, 2, 3 and that T r = k T r k is a homomorphism of algebras.In this paper, we first recognize the phenomenon that if we start from any degree d and apply Sq 0 repeatedly at most (k − 2) times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL k -representations. As a consequence, every finite Sq 0 -family in the coinvariants has at most (k − 2) nonzero elements. Two applications are exploited.The first main theorem is that T r k is not an isomorphism for k ≥ 5. Furthermore, for every k > 5, there are infinitely many degrees in which T r k is not an isomorphism. We also show that if T r detects a nonzero element in certain degrees of Ker(Sq 0 ), then it is not a monomorphism and further, for each k > , T r k is not a monomorphism in infinitely many degrees.The second main theorem is that the elements of any Sq 0 -family in the cohomology of the Steenrod algebra, except at most its first (k − 2) elements, are either all detected or all not detected by T r k , for every k. Applications of this study to the cases k = 4 and 5 show that T r 4 does not detect the three families g, D 3 and p , and that T r 5 does not detect the family {h n+1 g n | n ≥ 1}. algebraic transfer, mapping from the coinvariants of certain representations of the general linear group to the cohomology of the Steenrod algebra. Let V k denote a k-dimensional F 2 -vector space, and let P H * (BV k ) denote the primitive subspace consisting of all elements in H * (BV k ) that are annihilated by every positive-degree operation in the mod 2 Steenrod algebra, A. Throughout the paper, the homology is taken with coefficients in F 2 . The general linear group GL k := GL(V k ) acts regularly on V k and therefore on the homology and cohomology of BV k . Since the two actions of A and GL k upon H * (BV k ) commute with each other, there are inherited actions of GL k on F 2 ⊗ A Introduction and statement of results

2 ) be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that T r k is an isomorphism for k = 1, 2, 3. However, Singer showed that T r 5 is not an epimorphism. In this paper, we prove that T r 4 does not detect the nonzero element gs ∈ Ext 4,12·2 s A (F 2 , F 2 ) for every s ≥ 1. As a consequence, the localized (Sq 0 ) −1 T r 4 given by inverting the squaring operation Sq 0 is not an epimorphism. This gives a negative answer to a prediction by Minami.

Abstract. Let the mod 2 Steenrod algebra, A, and the general linear group,We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebraThis conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in Q 0 S 0 are the elements of Hopf invariant one and those of Kervaire invariant one.

This article contains a collection of problems contributed during the course of the conference. 55PXX; 55SXX1 The problems presented by Carles Broto Broto, Levi and Oliver [1,2] introduced the concept of p-local finite group as an algebraic object modeled on the fusion on the Sylow p-subgroup of a finite group, and attached to it a classifying space which is a p-complete space with properties similar to that of the p-completed classifying space of a finite group. Every finite group G gives rise canonically to a p-local finite group with classifying space homotopy equivalent to BG ∧ p . But there are also exotic examples; that is, p-local finite groups that cannot be obtained from a finite group in the canonic way and therefore its classifying space is not homotopy equivalent to the p-completed classifying space of any finite group.Some exotic examples are obtained by Broto and Møller [3] out of p-compact groups (see and Møller [7]). More precisely, to a 1-connected pcompact group X it is attached a family X(q) of p-local finite groups, q a p-adic unit, that approximates X in the sense that the classifying spaces {BX(q p m )} m∈N form a direct system with mapping telescope hocolim m BX(q p m ) BX

White Spot Syndrome Virus (WSSV) is the most damaging pathogen in terms of production and economic losses for the shrimp sector world-wide. Estimation of heritability for WSSV resistance was made in this study to obtain necessary parameter inputs for broadening the breeding objectives of an ongoing selective breeding programme for Whiteleg shrimp ( Liptopenaeus vannamei ) that has focussed exclusively on improving growth performance since 2014. The present study involved a disease challenge test experiment using a total of 15,000 shrimps from 150 full- and half-sib families (100 individuals per family). Survival rates were recorded at six different experimental periods: 1–3 days (S1), 1–5 days (S2), 1–7 days (S3), 1–9 days (S4), 1–12 days (S5), and 1–15 days (S6) and were used as measures of WSSV resistance. There was significant variation in WSSV resistance among families studied. Quantitative-real time PCR (qPCR) analysis showed that the amount of viral titer (viral load) was significantly lower in high than low resistance families. Analyses of heritability were carried out using linear mixed model (LMM) and threshold logistic generalized model (TLGM). Both linear and threshold models used showed that the heritability (h 2 ) for WSSV resistance was moderate in the early infection phases (S1–S4), whilst a low h 2 value was observed for survival after 12 and 15 days of the challenge test (S5 and S6). The transformed heritabilities for WSSV resistance ranged from 1 to 31% which were somewhat lower than those estimated on the liability scale. Genetic correlations between survival rates measured over six different days post-infection were high and positive (0.82–0.99). The phenotypic correlations ranged from 0.31 ± 0.01 to 0.97 ± 0.01. The genetic correlations between body weights and WSSV resistance were negative. Our results on the heritability and genetic correlations show that improvement of WSSV resistance can be achieved through selective breeding in this population of Whiteleg shrimp.