Steenrod problem
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[edit] 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 [Thom1954, Theorem III.4]. For every class in dimension of integral homology of a finite polyhedron K, there exists a non zero integer which depends only on p, such that the product is the image of the fundamental class of a closed oriented differentiable manifold.
More about that can be found in [Sullivan2004].
[edit] 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