We establish versions of the Snake Lemma from homological alge bra in the context of topological groups Banach spaces and operator algebras We apply this tool to demonstrate that if f B B is a quasi unital C map of separable C algebras so that it induces a map of Corona algebras f QB QB and if f is mono then the induced map f is also mono This paper presents a cross cultural result we use ideas...

In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3–manifolds, the unstable cohomology of the moduli space of curves and Faber’s conjecture, cokernel ...

In 1963 Atiyah and Singer proved the famous Atiyah-Singer Index Theorem, which states, among other things, that the space of elliptic pseudodifferential operators is such that the collection of operators with any given index forms a connected subset. Contained in this statement is the somewhat more specialized claim that the index of an elliptic operator must be invariant under sufficiently sma...

Let M1 n−1 denote the cokernel of the localization map BP∗/I → v−1 n−1BP∗/I, where I denotes the ideal of BP∗ generated by vi’s for 0 ≤ i ≤ n − 2. The chromatic Ext0(M1 n−1) on Γ(m + 1), which we denote Ex0(m, n), is isomorphic to the 0-th line of the E2-term of the Adams-Novikov spectral sequence for computing the homotopy groups of a spectrum, whose BP∗-homology is M1 n−1 ⊗BP∗ BP∗[t1, t2, . ....

For a finite, directed graph G = (V,E) we define the critical groupK(G) to be the cokernel of the transpose of the Laplacian matrix of G acting on Z , and K(G) to be its torsion subgroup. This generalizes the case of undirected graphs studied by Bacher, de la Harpe and Nagnibeda, and by Biggs. We prove a variety of results about these critical groups, among which are: that K(G/π) is a subgroup ...

We develop a topological approach to stratification learning. Given point cloud data drawn from a stratified space, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into dif...

For an infinite field $F$, we study the kernel of map $H_{n}(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]) \to H_{n}(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big])$ and cokernel $H_{n+1}\Big(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big) H_{n+1}\Big(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big)$. We give conjectural estimates these kernels c...

vanishes Note that index D M unlike the dimension of the ker nel and the dimension of the cokernel of D M is independent of the metric used in the construction of D M In fact according to the Atiyah Singer Index Theorem it is equal to a topological invariant A M the A genus ofM cf Ch III Thm We recall that A M is a characteristic number de ned by evaluating a certain polyno mial in the Pontrjag...

Let A be a supersingular abelian variety defined over a finite field k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f = > gi i for distinct monic irreducible polynomials gi and positive integers ei . We show that there is a group homomorphism .:...

Let Fn be a free group of rank n and F n the quotient group of Fn by the subgroup [Γn(3),Γn(3)][[Γn(2),Γn(2)],Γn(2)] where Γn(k) denotes the k-th subgroup of the lower central series of the free group Fn. In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n by using a generalized Chen’s integration in free groups. Then we apply it ...