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.
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