Steenrod problem
Encyclopedia
In mathematics, and particularly homology theory
, Steenrod's Problem (named after mathematician Norman Steenrod
) is a problem concerning the realisation of homology classes by singular manifolds.
, oriented manifold, and let be its orientation. Here denotes the n-dimensional homology group of M. Any continuous map defines an induced homomorphism
A homology class of Hn(X) is called realisable if it is of the form ƒ* [M] where The Steenrod problem is concerned with describing the realisable homology classes of Hn(X).
provided Moreover, any cycle can be realisable by the mapping of a pseudo-manifold.
The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of Hn(X,Z2), where Z2 denotes the integer
s modulo
2, can be realised by a non-oriented manifold
s MSO(k) clarified the Steenrod problem by reducing it to the study of the mappings A non-realisable class, , has been found where M is the Eilenberg–MacLane space: K(Z3⊕Z3,1).
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, Steenrod's Problem (named after mathematician Norman Steenrod
Norman Steenrod
Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:...
) is a problem concerning the realisation of homology classes by singular manifolds.
Formulation
Let M be a closedClosed manifold
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
, oriented manifold, and let be its orientation. Here denotes the n-dimensional homology group of M. Any continuous map defines an induced homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
A homology class of Hn(X) is called realisable if it is of the form ƒ* [M] where The Steenrod problem is concerned with describing the realisable homology classes of Hn(X).
Results
All elements of Hk(X) are realisable by smooth manifolds provided Any elements of Hn(X) are realisable by a mapping of a Poincaré complexPoincaré complex
In mathematics, and especially topology, a Poincaré complex is a generalisation of a manifold.The homology groups of a closed, orientable manifold are related in certain ways. This relationship came to be known as the Poincaré duality. Poincaré duality gives an isomorphism between homology and...
provided Moreover, any cycle can be realisable by the mapping of a pseudo-manifold.
The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of Hn(X,Z2), where Z2 denotes the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
2, can be realised by a non-oriented manifold
Conclusions
For smooth manifolds M the problem reduces to finding the form of the homomorphism , where Ωn(X) is the oriented bordism group of X. The connection between the bordisms Ω* and the Thom spaceThom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
s MSO(k) clarified the Steenrod problem by reducing it to the study of the mappings A non-realisable class, , has been found where M is the Eilenberg–MacLane space: K(Z3⊕Z3,1).