Calculating Euler-Poincare characteristic inductively
Geometric Topology
2022-03-22 v2 Combinatorics
Abstract
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 Euler-Poincare characteristic can be easily calculated without the use of any auxiliary combinatorial structure on spaces.
Keywords
Cite
@article{arxiv.1212.0154,
title = {Calculating Euler-Poincare characteristic inductively},
author = {Milosav M. Marjanovic},
journal= {arXiv preprint arXiv:1212.0154},
year = {2022}
}