Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X_0(Y_1)X_1 ... X_{n-1}(Y_n)X_n of a space Z. Among the applications is a succinct formula for the Euler-Poincare characteristic of Z, e(Z) = e(X_0) - e(Y_1) + e(X_1) - ... + e(X_{n-1}) - e(Y_n) + e(X_n) which exhibits the familiar sign pattern. A substantial part of the paper are examples demonstrating how the fibrous decomposition and consequently the… Expand

. In this paper we establish a topological property of geometric objects (lines, surfaces and solids) called Euler-Poincar´e characteristic. Since the paper is intended for a large proﬁle of… Expand

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility… Expand

This article surveys the Euler calculus - an integral calculus based on Euler characteristic - and its applications to data, sensing, networks, and imaging.

Over the 2016–2017 academic year, I ran the graduate algebraic topology sequence at MIT. The first semester traditionally deals with singular homology and cohomology and Poincaré duality; the second… Expand