duality between homotopy and homology of manifolds
I want to know if there IS a notion of DUALITY between the homotopy and homology of groups, namely topological manifolds given group structure. i.e, is there an equivalence between the number of simple closed curves or 'paths' that generate the space, and the number of connected components that generate the same given space.
I am reading Differential Forms in Algebraic Topology by Bott and Tuu, if anyone has insight into the book. I am fairly strong in my topology but not as strong in my algebra. THANKS!
differential-geometry algebraic-topology manifolds
asked Nov 27 '18 at 19:50
eyeheartmath
657
add a comment |
I want to know if there IS a notion of DUALITY between the homotopy and homology of groups, namely topological manifolds given group structure. i.e, is there an equivalence between the number of simple closed curves or 'paths' that generate the space, and the number of connected components that generate the same given space.
I am reading Differential Forms in Algebraic Topology by Bott and Tuu, if anyone has insight into the book. I am fairly strong in my topology but not as strong in my algebra. THANKS!
differential-geometry algebraic-topology manifolds
asked Nov 27 '18 at 19:50
eyeheartmath
657
2
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00
add a comment |
I want to know if there IS a notion of DUALITY between the homotopy and homology of groups, namely topological manifolds given group structure. i.e, is there an equivalence between the number of simple closed curves or 'paths' that generate the space, and the number of connected components that generate the same given space.
I am reading Differential Forms in Algebraic Topology by Bott and Tuu, if anyone has insight into the book. I am fairly strong in my topology but not as strong in my algebra. THANKS!
differential-geometry algebraic-topology manifolds
asked Nov 27 '18 at 19:50
eyeheartmath
657
I want to know if there IS a notion of DUALITY between the homotopy and homology of groups, namely topological manifolds given group structure. i.e, is there an equivalence between the number of simple closed curves or 'paths' that generate the space, and the number of connected components that generate the same given space.
I am reading Differential Forms in Algebraic Topology by Bott and Tuu, if anyone has insight into the book. I am fairly strong in my topology but not as strong in my algebra. THANKS!
differential-geometry algebraic-topology manifolds
differential-geometry algebraic-topology manifolds
asked Nov 27 '18 at 19:50
eyeheartmath
657
asked Nov 27 '18 at 19:50
eyeheartmath
657
asked Nov 27 '18 at 19:50
eyeheartmath
657
asked Nov 27 '18 at 19:50
eyeheartmath
657
asked Nov 27 '18 at 19:50
eyeheartmath
657
657
2
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00
add a comment |
2
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00
2
2
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00
add a comment |
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2
I'm not sure I understand the question. For example, what does it mean for a set of paths to generate a space?
– Jason DeVito
Nov 27 '18 at 21:04
Like Jason DeVito, I'm also unsure what you're asking. However, you might be interested in the Hurewicz homomorphism, which gives a relation between homotopy groups and homology groups. See: en.wikipedia.org/wiki/Hurewicz_theorem
– Jesse Madnick
Nov 27 '18 at 21:46
I should rephrase set of paths generating the space as follows: First homotopy group of S^1 is Z, so the group of paths in S^1 is isomorphic to Z as a group, group operation defined on paths (simple closed curves in R3), is there any notion of duality between this and the first homology group of S^1? I read in the book that there is a duality between path components and connected components of a given space. also I looked up Hurewicz homomorphism, and thanks SO much, thats what I meant. So there IS a duality between homology and homotopy of groups.
– eyeheartmath
Nov 28 '18 at 4:23
Duality is not the word you want to use here, I don't think. More like "relationship", or something more precise.
– Mike Miller
Nov 28 '18 at 8:00