Cubical cup poduct with a unit 1-cochain and the coboundary operator

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Thread starter CoreSpace
Start date Jul 30, 2024
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In summary, the study of the cubical cup product involves analyzing the interactions between unit 1-cochains and the coboundary operator within the context of algebraic topology. The cubical cup product provides a way to combine cochains, leading to the exploration of properties related to cohomology theories. The coboundary operator plays a crucial role in defining how these cochains relate to one another, facilitating the understanding of topological spaces through algebraic structures.

Jul 30, 2024

CoreSpace

TL;DR Summary: equivalence between cup product and coboundary operator

Define a u-coboundary operator u-d on a cubic p-cochain f as cup product multiplication U from the left with a unit 1-cochain: (u-d)f = 1Uf.
Because 1U1=0 and associativity of U, we have (u-d)^2=0.
What is the relation of u-d and the standard coboundary operator d?
Are they the same?