Simplicial Homology: Understanding & Adjoint Boundary Operator

  • Thread starter wofsy
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In summary: 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.
  • #1
wofsy
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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.
 
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  • #2
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.
 
  • #3
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.
 

FAQ: Simplicial Homology: Understanding & Adjoint Boundary Operator

1. What is simplicial homology?

Simplicial homology is a mathematical tool used to study the topological structure of a space. It is a way of assigning algebraic objects, such as groups, to a space in order to better understand its properties.

2. How is simplicial homology used in practice?

Simplicial homology is used in many areas of mathematics, including algebraic topology, differential geometry, and algebraic geometry. It is also used in applications such as image processing and data analysis.

3. What is the adjoint boundary operator in simplicial homology?

The adjoint boundary operator is a linear map that is used to define the boundaries of simplices in a simplicial complex. It is a key component in the calculation of homology groups, which are used to classify the topological properties of a space.

4. What is the significance of understanding simplicial homology?

Understanding simplicial homology allows for a deeper understanding of the topological structure of a space. This can lead to insights in other areas of mathematics and real-world applications, such as network analysis and data compression.

5. Are there any resources available for learning about simplicial homology?

Yes, there are many resources available for learning about simplicial homology, including textbooks, online courses, and video lectures. It is also helpful to have a strong understanding of linear algebra and basic topology before delving into simplicial homology.

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