Meeting Notes 2025-04-01

This Week’s Finds

No finds this week.

 

Focus Session

Speaker: Jim
Title: Chains, Boundaries, Cycles, Homology & Cohomology

 

1. Overview

• This session continued the homology series by defining chains, boundaries, cycles, and homology groups, then introduced cohomology analogues.

 

2. Main Content

1. Chains, Boundaries & Cycles
   • Chain groups \(C_n(X)\): free abelian groups generated by oriented \(n\)-simplices (or cells).
   • Boundary operator \(\partial_n: C_n(X)\to C_{n-1}(X)\) satisfying \(\partial_n\circ\partial_{n+1}=0\).
   • Cycle group \(Z_n(X)=\ker(\partial_n)\) and boundary group \(B_n(X)=\mathrm{im}(\partial_{n+1})\).
2. Homology Groups
   • Homology \(H_n(X)=Z_n(X)/B_n(X)\) measures equivalence classes of cycles modulo boundaries.
   • Examples:
     • \(H_0(X)\cong\mathbb{Z}^{\#\text{components}}\)
     • \(H_1(S^1)\cong\mathbb{Z}\)
   • Euler characteristic via ranks: \(\chi = \sum_n (-1)^n \mathrm{rank}\,H_n(X)\).
3. Cohomology
   • Cochain groups \(C^n(X)=\mathrm{Hom}(C_n(X),G)\) for coefficient group \(G\).
   • Coboundary operator \(\delta^n: C^n(X)\to C^{n+1}(X)\) with \(\delta^n\circ\delta^{n-1}=0\).
   • Cocycle group \(Z^n(X)=\ker(\delta^n)\) and coboundary group \(B^n(X)=\mathrm{im}(\delta^{n-1})\).
   • Cohomology \(H^n(X)=Z^n(X)/B^n(X)\), dual to homology.
   • Brief mention: Universal Coefficient Theorem relates \(H_n(X)\) and \(H^n(X)\).

 

3. Discussion Points

• Interpreting homology classes as equivalence classes of cycles.
• Comparing homology vs. cohomology: algebraic duality and ring structures.
• Computational considerations: working with chains vs. cochains.

 

Up Next

Next Talk:

• Alice on Cup Products & Cohomology Ring Structure

 

References

1. Hatcher, Algebraic Topology.
2. Munkres, Elements of Algebraic Topology.
3. Homology (Wikipedia)
4. Cohomology (Wikipedia)