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.[1]
Formulation
Let M be a closed, oriented manifold of dimension n, and let \( [M]\in H_{n}(M) \) be its orientation class. Here \( {\displaystyle H_{n}(M)} \) denotes the integral, n n-dimensional homology group of M. Any continuous map \( {\displaystyle f\colon M\to X} \) defines an induced homomorphism \( {\displaystyle f_{*}\colon H_{n}(M)\to H_{n}(X) \).[2] A homology class of \( H_n(X \)) is called realisable if it is of the form \( {\displaystyle f_{*}[M]} \) where \( [M]\in H_{n}(M) \). The Steenrod problem is concerned with describing the realisable homology classes of \( H_n(X \)).[3]
Results
All elements of \( H_{k}(X) \) are realisable by smooth manifolds provided \( {\displaystyle k\leq 6} \). Any elements of \( H_n(X \)) are realisable by a mapping of a Poincaré complex provided \( n\neq 3 \). Moreover, any cycle can be realized by the mapping of a pseudo-manifold.[3]
The assumption that M be orientable can be relaxed. In the case of non-orientable manifolds, every homology class of \( {\displaystyle H_{n}(X,\mathbb {Z} _{2})} \), where \( \mathbb{Z } _{2} \) denotes the integers modulo 2, can be realized by a non-oriented manifold, \( {\displaystyle f\colon M^{n}\to X} \).[3]
Conclusions
For smooth manifolds M the problem reduces to finding the form of the homomorphism \( {\displaystyle \Omega _{n}(X)\to H_{n}(X)} \), where \( {\displaystyle \Omega _{n}(X)} \) is the oriented bordism group of X.[4] The connection between the bordism groups \( {\displaystyle \Omega _{*}} \) and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms \( {\displaystyle H_{*}(\operatorname {MSO} (k))\to H_{*}(X)} \) .[3][5] In his landmark paper from 1954,[5] René Thom produced an example of a non-realisable class, \( {\displaystyle [M]\in H_{7}(X)} \), where M is the Eilenberg–MacLane space \( {\displaystyle K(\mathbb {Z} _{3}\oplus \mathbb {Z} _{3},1)} \).
See also
Singular homology
Pontryagin-Thom construction
Cobordism
References
Eilenberg, Samuel (1949). "On the problems of topology". Annals of Mathematics. 50: 247–260. doi:10.2307/1969448.
Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
Encyclopedia of Mathematics. "Steenrod Problem". Retrieved October 29, 2020.
Rudyak, Yuli B. (1987). "Realization of homology classes of PL-manifolds with singularities". Mathematical Notes. 41 (5): 417–421. doi:10.1007/bf01159869.
Thom, René (1954). "Quelques propriétés globales des variétés differentiable". Commentarii Mathematici Helvetici (in French). 28: 17–86. doi:10.1007/bf02566923.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
