Steenrod problem

Let $M$ be a closed, oriented manifold of dimension $n$, and let $[M]\; \backslash in\; H\_n(M)$ be its orientation class. Here $H\_n(M)$ denotes the integral, $n$-dimensional homology group of $M$. Any continuous map $f\backslash colon\; M\backslash to\; X$ defines an induced homomorphism $f\_*\backslash colon\; H\_n(M)\backslash to\; H\_n(X)$.{{Citation|first=Allen|last=Hatcher| authorlink=Allen Hatcher|title=Algebraic Topology|publisher=Cambridge University Press|year=2001|isbn=0-521-79540-0}} A homology class of $H\_n(X)$ is called realisable if it is of the form $f\_*[M]$ where $[M]\; \backslash in\; H\_n(M)$. The Steenrod problem is concerned with describing the realisable homology classes of $H\_n(X)$.{{Cite web|url=https://encyclopediaofmath.org/wiki/Steenrod_problem|author=Encyclopedia of Mathematics|title=Steenrod Problem|accessdate=October 29, 2020}}

All elements of $H\_k(X)$ are realisable by smooth manifolds provided $k\backslash le\; 6$. 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 $H\_n(X,\backslash Z\_2)$, where $\backslash Z\_2$ denotes the integers modulo 2, can be realized by a non-oriented manifold, $f\backslash colon\; M^n\backslash to\; X$.

For smooth manifolds M the problem reduces to finding the form of the homomorphism $\backslash Omega\_n(X)\; \backslash to\; H\_n(X)$, where $\backslash Omega\_n(X)$ is the oriented bordism group of X.{{Cite journal|last=Rudyak|first=Yuli B.|year=1987|title=Realization of homology classes of PL-manifolds with singularities|journal=Mathematical Notes|volume=41|issue=5|pages=417–421|doi=10.1007/bf01159869|s2cid=122228542 }} The connection between the bordism groups $\backslash Omega\_*$ and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms $H\_*(\backslash operatorname\{MSO\}(k))\; \backslash to\; H\_*(X)$.{{Cite journal|last=Thom|first=René |authorlink=René Thom|year=1954|title=Quelques propriétés globales des variétés differentiable|journal=Commentarii Mathematici Helvetici|volume=28|pages=17–86|language=French|doi=10.1007/bf02566923|s2cid=120243638 }} In his landmark paper from 1954, René Thom produced an example of a non-realisable class, $[M]\; \backslash in\; H\_7(X)$, where M is the Eilenberg–MacLane space $K(\backslash Z\_3\backslash oplus\; \backslash Z\_3,1)$.

• Singular homology
• Pontryagin-Thom construction
• Cobordism

{{Reflist}}

• [https://mathoverflow.net/q/48814 Thom construction and the Steenrod problem] on MathOverflow
• [https://mathoverflow.net/q/32828 Explanation for the Pontryagin-Thom construction]

Category:Homology theory

Category:Manifolds

Category:Geometric topology