Simplicial Homology: Understanding & Adjoint Boundary Operator

• wofsy
The identification at the chain level is more subtle - two cells are said to be homologous if they differ by a boundary if and only if the boundary operator at the chain level is the adjoint of the boundary operator at the homology level. In summary, the boundary operator works on an ordered simplex, simplices with the same vertices but different order are not identified, and the adjoint boundary operator is needed to get cohomology.f

wofsy

I need help understanding how simplicial homology works.

I understand how the boundary operator works on an ordered simplex. But how are simplices with the same vertices but different order identified? One can not say that they are the same if the the order determines the same orientation and negative if the orientations are opposite which is what I first thought. But then one seems to need degenerate simplices to get the right boundary relations. But i thought degenerate simplices were unnecessary.

Second, how does one define the adjoint boundary operator to get cohomology? This operator acts upon simplicial chains not on simplicial cochains.

I'm not sure that I see a case when you will get two sets of vertices appearing in an inconsistent order: you pick an orientation for each at the start and they should remain consistent, right? How about posting an example where this has happened in your calculations.

They aren't identified! For starters, the chain complex is a free abelian group, so different simplices (cells, etc.) in different positions (or with different orientations) are not identified. We don't even identify simplices (cells,etc.) that differ by a reparametrization. The original orientation is fixed on the standard n-simplex in Euclidean space, and the boundary operator is calculated using this specific orienation. Changing the orientation changes everything.

With that said, one could say that the cells that differ by an orientation are homologous if there exists a homeomorphism between their images (i.e. if there's an orientation-preserving automorphism of the standard n-simplex that gives you the prescribed orientation), or if they're homotopic, etc. But a distinction needs to be made between identification at the chain level (which we don't have), and identification at the homology level. The identification at the homology level is very explicit - two cells are homologous iff they differ by a boundary.