Meeting Notes 2025-02-25

This Week’s Finds

No finds this week.

 

Focus Session

Speaker: Jim
Title: Basic Structures in Homology: Simplex, Simplicial Complexes, and CW Complexes

 

1. Overview

• Introduced the combinatorial building blocks for defining homology groups.
• Emphasized the process of passing from geometric objects (simplices, cells) to algebraic invariants (chains, boundaries).

 

2. Main Content

1. Definition & Formalism
   • Simplex: The standard *n*-simplex \(\sigma^n = \mathrm{conv}\{v_0,\dots,v_n\}\), oriented by an ordering of its vertices; boundary operator \(\partial \sigma^n = \sum_{i=0}^n (-1)^i [v_0,\dots,\hat v_i,\dots,v_n]\).
   • Simplicial Complex: A collection \(K\) of simplices such that every face of a simplex in \(K\) is also in \(K\), and intersections of simplices are faces; geometric realization \(|K|\) carries the weak topology.
   • CW Complex: Built inductively via skeleta \(X^0 \subset X^1 \subset \cdots\), where \(X^n = X^{n-1} \cup \bigsqcup\text{ of }n\text{-cells}\) attached by maps \(S^{n-1}\to X^{n-1}\).
   • Skeletons: The *k*-skeleton \(K^{(k)}\) (or \(X^{(k)}\)) is the union of all cells (or simplices) of dimension ≤ *k*.
   • Combinatorial Properties:
     • f-vector $(f_0,\dots,f_n)$ counts the number of i-simplices.
     • Euler characteristic \(\chi = \sum_{i=0}^n (-1)^i f_i\).
     • Chain groups \(C_i(K)\), boundary maps \(\partial_i\) with \(\partial_i\circ\partial_{i+1}=0\).
2. Examples & Illustrations
   • 0-, 1-, 2-simplex diagrams with oriented edges and faces.
   • Triangulated \(S^1\): two 1-simplices \(\{v_0v_1, v_1v_0\}\) with vertices \(\{v_0,v_1\}\).
   • CW-structure on \(S^2\): one 0-cell and one 2-cell attached by the constant map.
   • Torus: shown both as a simplicial complex (triangulation) and as a CW complex (one 0-cell, two 1-cells, one 2-cell).
3. Historical Context
   • Poincaré (1895): first notions of “holes” via simplicial decompositions.
   • Eilenberg & Steenrod (1940s): axiomatic development of homology.
   • J. H. C. Whitehead (1949): introduction of CW complexes.
4. Mathematical Significance
   • Homology groups \(H_n(K)\) classify *n*-dimensional “holes.”
   • Betti numbers \(\beta_n = \mathrm{rank}\,H_n\) give coarse topological invariants.
   • Foundation for advanced tools: cohomology rings, spectral sequences, and persistent homology.

 

3. Discussion Points

• When is a CW decomposition preferable to a simplicial one, both theoretically and computationally?
• Trade-offs in computational homology: simplex count vs. cell-attachment complexity.
• Handling orientations and coefficient rings (\(\mathbb{Z}\) vs. \(\mathbb{Z}_2\)) in chain complexes.

 

Up Next

Next Talk:

• Alice on Cellular Homology & Cohomology Products

 

References

1. Hatcher, Algebraic Topology
2. Munkres, Elements of Algebraic Topology
3. Simplex (Wikipedia)
4. CW Complex (Wikipedia)