Mixing
Motivations for Homotopy Theory
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Guys we have motivations and significance behind every concept of Mathematics. As we know if two spaces are homeamorphic then they share lots of common properties - Euler's characteristics for instance. I know if two spaces are homotopically equivalent then they have same fundamental group. Apart from this, can anybody let me know which properties two homotopically equivalent spaces share? Or whey we even study this theory?
algebraic-topology
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
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add a comment |
$begingroup$
Guys we have motivations and significance behind every concept of Mathematics. As we know if two spaces are homeamorphic then they share lots of common properties - Euler's characteristics for instance. I know if two spaces are homotopically equivalent then they have same fundamental group. Apart from this, can anybody let me know which properties two homotopically equivalent spaces share? Or whey we even study this theory?
algebraic-topology
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
$endgroup$
1
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
1
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10
add a comment |
$begingroup$
Guys we have motivations and significance behind every concept of Mathematics. As we know if two spaces are homeamorphic then they share lots of common properties - Euler's characteristics for instance. I know if two spaces are homotopically equivalent then they have same fundamental group. Apart from this, can anybody let me know which properties two homotopically equivalent spaces share? Or whey we even study this theory?
algebraic-topology
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
$endgroup$
Guys we have motivations and significance behind every concept of Mathematics. As we know if two spaces are homeamorphic then they share lots of common properties - Euler's characteristics for instance. I know if two spaces are homotopically equivalent then they have same fundamental group. Apart from this, can anybody let me know which properties two homotopically equivalent spaces share? Or whey we even study this theory?
algebraic-topology
algebraic-topology
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
asked Jan 10 at 14:01
Prince ThomasPrince Thomas
615210
615210
1
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
1
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10
add a comment |
1
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
1
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10
1
1
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
1
1
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10
add a comment |
1 Answer
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oldest
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I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:
- Euler characteristic
- Fundamental group
- Higher homotopy groups
- All homology groups
- The cohomology ring
- ...
The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.
answered Jan 10 at 14:14
ChristophChristoph
12k1642
$endgroup$
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
add a comment |
Your Answer
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1 Answer
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$begingroup$
I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:
- Euler characteristic
- Fundamental group
- Higher homotopy groups
- All homology groups
- The cohomology ring
- ...
The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.
answered Jan 10 at 14:14
ChristophChristoph
12k1642
$endgroup$
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
add a comment |
$begingroup$
I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:
- Euler characteristic
- Fundamental group
- Higher homotopy groups
- All homology groups
- The cohomology ring
- ...
The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.
answered Jan 10 at 14:14
ChristophChristoph
12k1642
$endgroup$
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
add a comment |
$begingroup$
I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:
- Euler characteristic
- Fundamental group
- Higher homotopy groups
- All homology groups
- The cohomology ring
- ...
The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.
answered Jan 10 at 14:14
ChristophChristoph
12k1642
$endgroup$
I'd argue that most topological invariants are actually homotopy invariants. Obviously "homeomorphism type" is not, but here is a small list of things that are homotopy invariants:
- Euler characteristic
- Fundamental group
- Higher homotopy groups
- All homology groups
- The cohomology ring
- ...
The idea behind all these invariants is to differentiate or even classify topological spaces. Hence, calculating these invariants is an important task and since they are homotopy invariants it is often easier to exchange your space $X$ for a homotopy equivalent space $Y$ and then calculate the invariants.
answered Jan 10 at 14:14
ChristophChristoph
12k1642
answered Jan 10 at 14:14
ChristophChristoph
12k1642
answered Jan 10 at 14:14
ChristophChristoph
12k1642
answered Jan 10 at 14:14
ChristophChristoph
12k1642
12k1642
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
add a comment |
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
2
2
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I'd argue that most algebraic topological invariants are actually homotopy invariants. Compactness is not a homotopy invariant but is an important topological invariant. Dimension also, if we're talking about manifolds.
$endgroup$
– John Palmieri
Jan 10 at 16:23
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
I agree! (I guess I tend to be more on the algebraic than the point set side of things, so this answer might be biased.)
$endgroup$
– Christoph
Jan 10 at 18:31
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
$begingroup$
Oh, I'm very algebraic myself, so I have the same biases, but we should be explicit about them. ;)
$endgroup$
– John Palmieri
Jan 10 at 20:55
add a comment |
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1
$begingroup$
Euler characteristic is an example.
$endgroup$
– Berci
Jan 10 at 14:02
1
$begingroup$
All higher homotopy groups, all homology groups, the cohomology ring, ...
$endgroup$
– Christoph
Jan 10 at 14:10