Homology groups with different complexes
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When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes.
Does it matter which one I use in general? Do they always yield the same homology groups?
Thank you!
algebraic-topology homology-cohomology cw-complexes
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
add a comment |
up vote
2
down vote
favorite
When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes.
Does it matter which one I use in general? Do they always yield the same homology groups?
Thank you!
algebraic-topology homology-cohomology cw-complexes
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
1
Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
1
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes.
Does it matter which one I use in general? Do they always yield the same homology groups?
Thank you!
algebraic-topology homology-cohomology cw-complexes
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
When computing homology groups, I have seen some people use simplicial complexes, delta complexes or CW-complexes.
Does it matter which one I use in general? Do they always yield the same homology groups?
Thank you!
algebraic-topology homology-cohomology cw-complexes
algebraic-topology homology-cohomology cw-complexes
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
asked Dec 2 at 4:31
MathUser_NotPrime
1,113112
1,113112
1
Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
1
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42
add a comment |
1
Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
1
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42
1
1
Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
1
1
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42
add a comment |
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Yes. All three satisfy the The Eilenberg–Steenrod axioms for homology.
– Matematleta
Dec 2 at 4:40
1
@Matematleta Thanks! Then do you know why most sources use delta complexes?
– MathUser_NotPrime
Dec 2 at 4:45
It depends on what you want to compute I guess. It's good to have all of them in the arsenal. See here for example.
– Matematleta
Dec 2 at 4:52
It doesn't matter anything and in fact, they all satisfy the E-S axioms and be isomorphic to integral homology. The problem depends on what kind of space you're trying to compute its invariant, something is solid then use simplicial homology while you should use cw homology for something's soft.
– bangbang1412
Dec 2 at 9:42