Steenrod problem
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The answer was given by Thom in 1954 {{cite|Thom1954}}. He showed that in dimensions $0 \leq m \leq 6$ this is true but in general this is not the case. He constructed a counter example in dimension $m=7$. Thom also showed that the corresponding problem with $\mathbb{Z}/2$ coefficients is correct, that is the corresponding homomorphism $\Phi : \Omega^{O}_{∗}(X) \to H_{*}(X,\mathbb{Z}/2)$ is surjective. | The answer was given by Thom in 1954 {{cite|Thom1954}}. He showed that in dimensions $0 \leq m \leq 6$ this is true but in general this is not the case. He constructed a counter example in dimension $m=7$. Thom also showed that the corresponding problem with $\mathbb{Z}/2$ coefficients is correct, that is the corresponding homomorphism $\Phi : \Omega^{O}_{∗}(X) \to H_{*}(X,\mathbb{Z}/2)$ is surjective. | ||
+ | Thom also proved the following: | ||
+ | {{beginthm|Theorem 1|(Thom)}} | ||
+ | For every class in dimension $p$ of integral homology of a finite polyhedron K, there exists a non zero integer $N$, such that the product $Nz$ is the image of a fundamental class of a closed oriented differentiable manifold. | ||
+ | {{endthm}}. | ||
+ | |||
+ | More about that can be found in {{cite|Sullivan2004}} | ||
</wikitex> | </wikitex> | ||
Revision as of 17:08, 31 March 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Tex syntax error. The elements in the image of are called representable.
In certain situations it is convenient to assume that a homology class is representable. In dimensions and it is clear that is surjective (even an isomorphism). It is less obvious in dimension , but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 [Eilenberg1949]:
Given a simplicial complex , is every (integral) homology class representable?
The answer was given by Thom in 1954 [Thom1954]. He showed that in dimensions this is true but in general this is not the case. He constructed a counter example in dimension . Thom also showed that the corresponding problem with coefficients is correct, that is the corresponding homomorphism is surjective. Thom also proved the following:
Theorem 1 1.1 (Thom). For every class in dimension of integral homology of a finite polyhedron K, there exists a non zero integer , such that the product is the image of a fundamental class of a closed oriented differentiable manifold.
More about that can be found in [Sullivan2004]
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Sullivan2004] D. Sullivan, René Thom's work on geometric homology and bordism, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no.3, 341–350 (electronic). MR2058291 Zbl 1045.57001
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
Tex syntax error. The elements in the image of are called representable.
In certain situations it is convenient to assume that a homology class is representable. In dimensions and it is clear that is surjective (even an isomorphism). It is less obvious in dimension , but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 [Eilenberg1949]:
Given a simplicial complex , is every (integral) homology class representable?
The answer was given by Thom in 1954 [Thom1954]. He showed that in dimensions this is true but in general this is not the case. He constructed a counter example in dimension . Thom also showed that the corresponding problem with coefficients is correct, that is the corresponding homomorphism is surjective. Thom also proved the following:
Theorem 1 1.1 (Thom). For every class in dimension of integral homology of a finite polyhedron K, there exists a non zero integer , such that the product is the image of a fundamental class of a closed oriented differentiable manifold.
More about that can be found in [Sullivan2004]
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Sullivan2004] D. Sullivan, René Thom's work on geometric homology and bordism, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no.3, 341–350 (electronic). MR2058291 Zbl 1045.57001
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
Tex syntax error. The elements in the image of are called representable.
In certain situations it is convenient to assume that a homology class is representable. In dimensions and it is clear that is surjective (even an isomorphism). It is less obvious in dimension , but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 [Eilenberg1949]:
Given a simplicial complex , is every (integral) homology class representable?
The answer was given by Thom in 1954 [Thom1954]. He showed that in dimensions this is true but in general this is not the case. He constructed a counter example in dimension . Thom also showed that the corresponding problem with coefficients is correct, that is the corresponding homomorphism is surjective. Thom also proved the following:
Theorem 1 1.1 (Thom). For every class in dimension of integral homology of a finite polyhedron K, there exists a non zero integer , such that the product is the image of a fundamental class of a closed oriented differentiable manifold.
More about that can be found in [Sullivan2004]
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Sullivan2004] D. Sullivan, René Thom's work on geometric homology and bordism, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no.3, 341–350 (electronic). MR2058291 Zbl 1045.57001
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
Tex syntax error. The elements in the image of are called representable.
In certain situations it is convenient to assume that a homology class is representable. In dimensions and it is clear that is surjective (even an isomorphism). It is less obvious in dimension , but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 [Eilenberg1949]:
Given a simplicial complex , is every (integral) homology class representable?
The answer was given by Thom in 1954 [Thom1954]. He showed that in dimensions this is true but in general this is not the case. He constructed a counter example in dimension . Thom also showed that the corresponding problem with coefficients is correct, that is the corresponding homomorphism is surjective. Thom also proved the following:
Theorem 1 1.1 (Thom). For every class in dimension of integral homology of a finite polyhedron K, there exists a non zero integer , such that the product is the image of a fundamental class of a closed oriented differentiable manifold.
More about that can be found in [Sullivan2004]
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Sullivan2004] D. Sullivan, René Thom's work on geometric homology and bordism, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no.3, 341–350 (electronic). MR2058291 Zbl 1045.57001
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502