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.

Formulation

Let M {\displaystyle M} be a closed, oriented manifold of dimension n {\displaystyle n}, and let [ M ] ∈ H n ( M ) {\displaystyle [M]\in H_{n}(M)} be its orientation class. Here H n ( M ) {\displaystyle H_{n}(M)} denotes the integral, n {\displaystyle n}-dimensional homology group of M {\displaystyle M}. Any continuous map f : M → X {\displaystyle f\colon M\to X} defines an induced homomorphism f ∗ : H n ( M ) → H n ( X ) {\displaystyle f_{*}\colon H_{n}(M)\to H_{n}(X)}. A homology class of H n ( X ) {\displaystyle H_{n}(X)} is called realisable if it is of the form f ∗ [ M ] {\displaystyle f_{*}[M]} for some manifold M {\displaystyle M} and map f : M → X {\displaystyle f:M\to X}. The Steenrod problem is concerned with describing the realisable homology classes of H n ( X ) {\displaystyle H_{n}(X)}.

Results

All elements of H k ( X ) {\displaystyle H_{k}(X)} are realisable by smooth manifolds provided k ≤ 6 {\displaystyle k\leq 6}. Moreover, any cycle can be realized 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 H n ( X , Z 2 ) {\displaystyle H_{n}(X,\mathbb {Z} _{2})}, where Z 2 {\displaystyle \mathbb {Z} _{2}} denotes the integers modulo 2, can be realized by a non-oriented manifold, f : M n → X {\displaystyle f\colon M^{n}\to X}.

Conclusions

For smooth manifolds M the problem reduces to finding the form of the homomorphism Ω n ( X ) → H n ( X ) {\displaystyle \Omega _{n}(X)\to H_{n}(X)}, where Ω n ( X ) {\displaystyle \Omega _{n}(X)} is the oriented bordism group of X. The connection between the bordism groups Ω ∗ {\displaystyle \Omega _{*}} and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms H ∗ ( MSO ⁡ ( k ) ) → H ∗ ( X ) {\displaystyle H_{*}(\operatorname {MSO} (k))\to H_{*}(X)}. In his landmark paper from 1954, René Thom produced an example of a non-realisable class, [ M ] ∈ H 7 ( X ) {\displaystyle [M]\in H_{7}(X)}, where M is the Eilenberg–MacLane space K ( Z 3 ⊕ Z 3 , 1 ) {\displaystyle K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)}.

See also

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