Steenrod problem
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Given a space X, there is a homomorphism $\Phi : \Omega^{SO}_{∗}(X) \to H_{*}(X,\mathbb{Z})$, called the Thom homomorphism, given by $[M,f] \to f_{*}([M])$ where $[M]$ is the fundamental class of $M$. The elements in the image of $\Phi$ are called representable. | Given a space X, there is a homomorphism $\Phi : \Omega^{SO}_{∗}(X) \to H_{*}(X,\mathbb{Z})$, called the Thom homomorphism, given by $[M,f] \to f_{*}([M])$ where $[M]$ is the fundamental class of $M$. The elements in the image of $\Phi$ are called representable. | ||
In certain situations it is convenient to assume that a homology class is representable. In dimensions $0$ and $1$ it is clear that $\Phi$ is surjective (even an isomorphism). It is less obvious in dimension $2$, but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 {{cite|Eilenberg1949}}: | In certain situations it is convenient to assume that a homology class is representable. In dimensions $0$ and $1$ it is clear that $\Phi$ is surjective (even an isomorphism). It is less obvious in dimension $2$, but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 {{cite|Eilenberg1949}}: | ||
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Given a simplicial complex $X$, is every (integral) homology class representable? | Given a simplicial complex $X$, is every (integral) homology class representable? | ||
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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. | ||
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Revision as of 12:51, 31 March 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Introduction
Given a space X, there is a homomorphism , called the Thom homomorphism, given by where is the fundamental class of . 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.
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
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.
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502
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.
2 References
- [Eilenberg1949] S. Eilenberg, On the problems of topology, Ann. of Math. (2) 50 (1949), 247–260. MR0030189 (10,726b) Zbl 0034.25304
- [Thom1954] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86. MR0061823 (15,890a) Zbl 0057.15502