Adams spectral sequence and higher torsion in MSp

In this paper we study higher torsion in the symplectic cobordism ring. We use Toda brackets and manifolds with singularities to construct elements of higher torsion and use the Adams spectral sequence to determine an upper bound for the order of these elements

Infinite loop spaces and positive scalar curvature

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real $K$-theory spectrum. Our main results concern the nontriviality of this map. We prove that for $2n \geq 6$, the natural $KO$-orientation from the infinite loop space of the Madsen--Tillmann--Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any $2n$-dimensional spin manifold. For manifolds of odd dimension $2n+1 \geq 7$, we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov--Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin's secondary index to several other index-theoretical results, such as the Atiyah--Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral-flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen--Weiss theorem, proven by Galatius and the third named author.Johannes Ebert was partially supported by the SFB 878. O. Randal-Williams acknowledges Herchel Smith Fellowship support

GOATOOLS: A Python library for Gene Ontology analyses.

The biological interpretation of gene lists with interesting shared properties, such as up- or down-regulation in a particular experiment, is typically accomplished using gene ontology enrichment analysis tools. Given a list of genes, a gene ontology (GO) enrichment analysis may return hundreds of statistically significant GO results in a "flat" list, which can be challenging to summarize. It can also be difficult to keep pace with rapidly expanding biological knowledge, which often results in daily changes to any of the over 47,000 gene ontologies that describe biological knowledge. GOATOOLS, a Python-based library, makes it more efficient to stay current with the latest ontologies and annotations, perform gene ontology enrichment analyses to determine over- and under-represented terms, and organize results for greater clarity and easier interpretation using a novel GOATOOLS GO grouping method. We performed functional analyses on both stochastic simulation data and real data from a published RNA-seq study to compare the enrichment results from GOATOOLS to two other popular tools: DAVID and GOstats. GOATOOLS is freely available through GitHub: https://github.com/tanghaibao/goatools

GOATOOLS: A Python library for Gene Ontology analyses

The biological interpretation of gene lists with interesting shared properties, such as up- or down-regulation in a particular experiment, is typically accomplished using gene ontology enrichment analysis tools. Given a list of genes, a gene ontology (GO) enrichment analysis may return hundreds of statistically significant GO results in a “flat” list, which can be challenging to summarize. It can also be difficult to keep pace with rapidly expanding biological knowledge, which often results in daily changes to any of the over 47,000 gene ontologies that describe biological knowledge. GOATOOLS, a Python-based library, makes it more efficient to stay current with the latest ontologies and annotations, perform gene ontology enrichment analyses to determine over- and under-represented terms, and organize results for greater clarity and easier interpretation using a novel GOATOOLS GO grouping method. We performed functional analyses on both stochastic simulation data and real data from a published RNA-seq study to compare the enrichment results from GOATOOLS to two other popular tools: DAVID and GOstats. GOATOOLS is freely available through GitHub: https://github.com/tanghaibao/goatools

Adams Spectral Sequence and Higher Torsion in MSp*

In a previous paper, we constructed higher torsion elements in the symplectic cobordism ring and used chromatic computations in the Adams-- Novikov spectral sequence to obtain a lower bound for their order. In this paper we use the Adams spectral sequence to depict our understanding of the nature of higher torsion elements in the symplectic cobordism ring. We use Toda brackets and manifolds with singularities to improve upon our previous constructions and use higher differentials in the Adams spectral sequence to determine an upper bound for the order of these elements. 1 Introduction The symplectic cobordism ring MSp is the homotopy of the Thom spectrum MSp and classifies up to cobordism the ring of smooth manifolds with a symplectic structure on their stable normal bundles. Although MSp only has two-- torsion, its ring structure is very complicated and is only completely understood through the 100 stem [5], [11]. In [2], we constructed higher two torsion elements of all orders in..

Singularities and Higher Torsion in Symplectic Cobordism I

In this paper we construct higher two--torsion elements of all orders in the symplectic cobordism ring. We begin by constructing higher torsion elements in the symplectic cobordism ring with singularities using a geometric approach to the Adams-Novikov spectral sequence in terms of cobordism with singularities. Then we show how these elements determine particular elements of higher torsion in the symplectic cobordism ring. 1 Introduction The symplectic cobordism ring MSp is the homotopy of the Thom spectrum MSp and classifies up to cobordism the ring of smooth manifolds with an Sp- structure on their stable normal bundles. Although MSp only has two--torsion, its ring structure is far more complicated than any of the other cobordism rings MG for the classical Lie groups G = O, SO, Spin, U , SU which have been completely computed. Over the past thirty years, these cobordism rings MG have had a major impact on differential topology and homotopy theory. On the other hand, if the c..