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Difference between revisions of "Steenrod problem"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Eilenberg,   "On the problems of topology" ''Ann. of Math.'' , '''50''' (1949) pp. 247–260</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Thom,   "Quelques propriétés globales des variétés differentiables" ''Comm. Math. Helv.'' , '''28''' (1954) pp. 17–86</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.B. Rudyak,   "Realization of homology classes of PL-manifolds with singularities" ''Math. Notes'' , '''41''' : 5 (1987) pp. 417–421 ''Mat. Zametki'' , '''41''' : 5 (1987) pp. 741–749</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Eilenberg, "On the problems of topology" ''Ann. of Math.'' , '''50''' (1949) pp. 247–260 {{MR|0030189}} {{ZBL|0034.25304}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Thom, "Quelques propriétés globales des variétés differentiables" ''Comm. Math. Helv.'' , '''28''' (1954) pp. 17–86 {{MR|0061823}} {{ZBL|0057.15502}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) {{MR|0176478}} {{ZBL|0125.40103}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) {{MR|0248858}} {{ZBL|0181.26604}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.B. Rudyak, "Realization of homology classes of PL-manifolds with singularities" ''Math. Notes'' , '''41''' : 5 (1987) pp. 417–421 ''Mat. Zametki'' , '''41''' : 5 (1987) pp. 741–749 {{MR|898135}} {{ZBL|0632.57020}} </TD></TR></table>

Revision as of 17:01, 15 April 2012

The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. [1]. Let be a closed oriented manifold (topological, piecewise-linear, smooth, etc.) and let be its orientation (here is the -dimensional homology group of ). Any continuous mapping defines an element . The Steenrod problem consists of describing those homology classes of , called realizable, which are obtained in this way, i.e. which take the form for a certain from the given class. All elements of the groups , , are realizable by a smooth manifold. Any element of the group , , is realizable by a mapping of a Poincaré complex . Moreover, any cycle can be realized by a pseudo-manifold. Non-orientable manifolds can also be considered, and every homology class modulo (i.e. element of ) can be realized by a non-oriented smooth singular manifold .

Thus, for smooth the Steenrod problem consists of describing the form of the homomorphism , where is the oriented bordism group of the space. The connection between the bordisms and the Thom spaces (cf. Thom space) , discovered by R. Thom [2], clarified the Steenrod problem by reducing it to the study of the mappings . A non-realizable class has been exhibited, where is the Eilenberg–MacLane space . For any class , some multiple is realizable (by a smooth manifold); moreover, can be chosen odd.

References

[1] S. Eilenberg, "On the problems of topology" Ann. of Math. , 50 (1949) pp. 247–260 MR0030189 Zbl 0034.25304
[2] R. Thom, "Quelques propriétés globales des variétés differentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86 MR0061823 Zbl 0057.15502
[3] P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[4] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) MR0248858 Zbl 0181.26604
[5] Yu.B. Rudyak, "Realization of homology classes of PL-manifolds with singularities" Math. Notes , 41 : 5 (1987) pp. 417–421 Mat. Zametki , 41 : 5 (1987) pp. 741–749 MR898135 Zbl 0632.57020
How to Cite This Entry:
Steenrod problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_problem&oldid=16033
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article