Hopf Algebras and Invariants of the Johnson Cokernel

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F_n) on the nth tensor power of H which induces an Out(F_n) action on a quotient \overline{H^{\otimes n}}. In the case when H=T(V) is the tensor algebra, we show that the invariant Tr^C of the cokernel of the Johnson homomorphism studied in [J. Conant, The Johnson cokernel and the Enomoto-Satoh invariant, Algebraic and Geometric Topology, 15 (2015), no. 2, 801--821.] projects to take values in the top dimensional cohomology of Out(F_n) with coefficients in \overline{H^{\otimes n}}. We analyze the n=2 case, getting large families of obstructions generalizing the abelianization obstructions of [J. Conant, M. Kassabov, K. Vogtmann, Higher hairy graph homology, Journal of Topology, Geom. Dedicata 176 (2015), 345--374.]

Grope cobordism of classical knots

We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants. An interesting refinement we study are knots modulo symmetric grope cobordism in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot concordance group, via symmetric grope cobordism in 4-space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at h=2, Casson-Gordon invariants at h=3, and for h=4 one has the new von Neumann signatures of a knot.Comment: Final version. To appear in Topology. See http://www.math.cornell.edu/~jconant/pagethree.html for a PDF file with better figure qualit

Higher Order Intersections in Low-Dimensional Topology

We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants like Milnor, Sato-Levine and Arf invariants. We also define higher- order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the non- triviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described. This article is an announcement and summary of results to be published in several forthcoming papers

Two-loop part of the rational homotopy of spaces of long embeddings

Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop part.Comment: 19 pages, 2 figures. (No changes with previous version

Discrete Homotopy Theory and Critical Values of Metric Spaces

Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3/2. The latter two spectra are known to be discrete for compact geodesic spaces, and correspond to the values at which certain special covering maps, called delta-covers (Sormani-Wei) or epsilon-covers (Plaut-Wilkins), change equivalence type. In this paper we initiate the study of these ideas for non-geodesic spaces, motivated by the need to understand the extent to which the accompanying covering maps are topological invariants. We show that discreteness of the critical spectrum for general metric spaces can fail in several ways, which we classify. The "newcomer" critical values for compact, non-geodesic spaces are completely determined by the homotopy critical values and refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.Comment: 5 figures, 23 pages. This third version includes updated references, additions to the introduction that further motivate the investigation of the critical spectrum for non-geodesic spaces, and an answer to a question posed by the authors in the first version regarding the topological relevance of refinement critical value

Genome-wide association for milk production and lactation curve parameters in Holstein dairy cows

The aim of this study was to identify genomic regions associated with 305-day milk yield and lactation curve parameters on primiparous (n = 9,910) and multiparous (n = 11,158) Holstein cows. The SNP solutions were estimated using a weighted single-step genomic BLUP approach and imputed high-density panel (777k) genotypes. The proportion of genetic variance explained by windows of 50 consecutive SNP (with an average of 165 Kb) was calculated, and regions that accounted for more than 0.50% of the variance were used to search for candidate genes. Estimated heritabilities were 0.37, 0.34, 0.17, 0.12, 0.30 and 0.19, respectively, for 305-day milk yield, peak yield, peak time, ramp, scale and decay for primiparous cows. Genetic correlations of 305-day milk yield with peak yield, peak time, ramp, scale and decay in primiparous cows were 0.99, 0.63, 0.20, 0.97 and -0.52, respectively. The results identified three windows on BTA14 associated with 305-day milk yield and the parameters of lactation curve in primi- and multiparous cows. Previously proposed candidate genes for milk yield supported by this work include GRINA, CYHR1, FOXH1, TONSL, PPP1R16A, ARHGAP39, MAF1, OPLAH and MROH1, whereas newly identified candidate genes are MIR2308, ZNF7, ZNF34, SLURP1, MAFA and KIFC2 (BTA14). The protein lipidation biological process term, which plays a key role in controlling protein localization and function, was identified as the most important term enriched by the identified genes