In this paper, we take a topological view of unsupervised learning. From point view, clustering may be interpreted as trying to find the number connected components any underlying geometrically structured probability distribution in certain sense that will make precise. We construct seems appropriate for modeling data very high dimensions. A special case our construction is mixture Gaussians where there Gaussian noise concentrated around finite set points (the means). More generally consider...

The object of this paper is to begin a geometric study noncompact spaces whose local structure has bounded complexity. Manifolds sort arise as leaves foliations compact manifolds and their universal covers. We shall introduce coarse homology theory using chains complexity some its first properties. most interesting result characterizes when H uf (X) vanishes an analogue strengthening F0lner's amenability criterion for groups in terms isoperimetric inequalities. (See [4].) One can view...

Part 1 Manifold theory: algebraic K-theory and topology surgery theory spacification functoriality applications. 2 General definitions examples classification of stratified spaces transverse PT category controlled proof main theorems in topology. 3 Applications illustrations: manifolds embedding revisited supernormal varieties group actions rigidity conjectures.

We construct examples of nonresolvable generalized n-manifolds, n ≥ 6, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed n-manifold.We further investigate the structure manifolds and present a program for understanding their topology.

In this paper, we study lower bounds on the K‐theory of maximal C ‐algebra a discrete group based amount torsion it contains. We call finite part operator and give bound that is valid for large class groups, called finitely embeddable groups. The groups includes all residually amenable Gromov’s monster virtually torsion-free (eg Out.Fn/), any analytic diffeomorphisms an connected manifold fixing given point. apply result to measure degree nonrigidity compact oriented M with dimension 4k 1 .k...

We compute the homology of random Čech complexes over a homogeneous Poisson process on d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to Erdős -Rényi transition, where complex becomes connected. second all other groups are computed correctly (almost simultaneously). Our calculations also suggest finer measurement scales, further refinement this picture separation between different groups. © 2016 Wiley Periodicals, Inc. Random...

Abstract We show that the rational Novikov conjecture for a group Γ of finite homological type follows from mod 2 acyclicity Higson compactification an EΓ. then groups asymptotic dimension, is p acyclic all and deduce integral these groups. © 2007 Wiley Periodicals, Inc.

For each k ∈ Z, we construct a uniformly contractible metric on Euclidean space which is not mod hypereuclidean.We also pair of Riemannian metrics R n , ≥ 11, so that the resulting manifolds Z and are bounded homotopy equivalent by equivalence boundedly close to homeomorphism.We show for these spaces C * -algebra assembly map K lf (Z) → (C (Z)) from locally finite K-homology K-theory propagation algebra monomorphism.This shows an integral version coarse Novikov conjecture fails real operator...

We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(M) not torsion-free, then there are infinitely many manifolds homotopic equivalent to but homeomorphic it. To show the infinite size structure set M, we construct secondary invariant tau_(2): S(M)-->R coincides with rho-invariant Cheeger-Gromov. In particular, our result shows homotopy for in question.

Abstract Let X be a closed oriented connected topological manifold of dimension n ≥ 5 . The structure group is the abelian equivalence classes all pairs ( f , M ) such that and : → an orientation‐preserving homotopy equivalence. main purpose this article to prove higher rho invariant map defines homomorphism from analytic Here universal cover Γ = π 1 fundamental certain C * ‐algebra. In fact, we introduce on homology manifold, its additivity. This restricts group. More generally, same...

Abstract We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\textbf{Diff}$ or $\textbf{PL}$ is the of whether $m$-dimensional $CAT$-manifold immersible in $\mathbb{R}^{n}$ decidable? show that PL immersibility decidable all cases except codimension 2, whereas smooth odd codimensions undecidable many even codimensions. As a corollary, we embeddability $m$-manifold with boundary when $n-m$ $11m \geq 10n+1$.

Intersection homology and results related to the higher signature problem are applied show that certain combinations of eta-invariants operator homotopy invariant in various circumstances.

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers have a nontrivial amount symmetry. By we mean that Isom(M) not discrete. the well-known theorem Myers-Steenrod [MS], condition equivalent [Isom(M) : π1(M)] = ∞. Also note if any cover has nondiscrete isometry group, then so does its . Our description such given in Theorem 1.2 below. proof uses methods from Lie theory, harmonic maps, large-scale geometry, and homological theory...

Abstract Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety external conditions, which viewed parameters form part the input underlying problem. In this paper, we analyze parametrized problem for many distinct points in plane, moving without collision avoiding multiple obstacles with priori unknown positions. This complements our prior Cohen et al. [3] (SIAM J. Appl. Algebra Geom. 5 , 229–249), where were...