Steenrod problem
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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.
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
![[M,f] \to f_{*}([M])](/images/math/c/7/1/c711fc19bedc6bd184836dcdf7eb3273.png)
![[M]](/images/math/f/a/0/fa08c3d5d2f54260952acc8a646b5025.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![0](/images/math/d/0/a/d0a87271a40bebf0cd626354a0c0aee2.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
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
![[M,f] \to f_{*}([M])](/images/math/c/7/1/c711fc19bedc6bd184836dcdf7eb3273.png)
![[M]](/images/math/f/a/0/fa08c3d5d2f54260952acc8a646b5025.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![0](/images/math/d/0/a/d0a87271a40bebf0cd626354a0c0aee2.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
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
![[M,f] \to f_{*}([M])](/images/math/c/7/1/c711fc19bedc6bd184836dcdf7eb3273.png)
![[M]](/images/math/f/a/0/fa08c3d5d2f54260952acc8a646b5025.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![0](/images/math/d/0/a/d0a87271a40bebf0cd626354a0c0aee2.png)
![1](/images/math/0/6/d/06d3730efc83058f497d3d44f2f364e3.png)
![\Phi](/images/math/1/c/3/1c3cd16fb629f8676e01c67f501279d6.png)
![2](/images/math/0/3/3/0339b8cd4613c74a86d715439a3c09f7.png)
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