View source for Steenrod problem

The problem of the realization of cycles (homology classes) by singular manifolds; formulated by N. Steenrod, cf. [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875401.png" /> be a closed oriented manifold (topological, piecewise-linear, smooth, etc.) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875402.png" /> be its orientation (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875403.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875404.png" />-dimensional [[Homology group|homology group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875405.png" />). Any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875406.png" /> defines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875407.png" />. The Steenrod problem consists of describing those homology classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875408.png" />, called realizable, which are obtained in this way, i.e. which take the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s0875409.png" /> for a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754010.png" /> from the given class. All elements of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754012.png" />, are realizable by a smooth manifold. Any element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754014.png" />, is realizable by a mapping of a [[Poincaré complex|Poincaré complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754015.png" />. Moreover, any cycle can be realized by a [[Pseudo-manifold|pseudo-manifold]]. Non-orientable manifolds can also be considered, and every homology class modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754016.png" /> (i.e. element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754017.png" />) can be realized by a non-oriented smooth singular manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754018.png" />.

Thus, for smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754019.png" /> the Steenrod problem consists of describing the form of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754021.png" /> is the oriented [[Bordism|bordism]] group of the space. The connection between the bordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754022.png" /> and the Thom spaces (cf. [[Thom space|Thom space]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754023.png" />, discovered by R. Thom [[#References|[2]]], clarified the Steenrod problem by reducing it to the study of the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754024.png" />. A non-realizable class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754025.png" /> has been exhibited, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754026.png" /> is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754027.png" />. For any class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754028.png" />, some multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754029.png" /> is realizable (by a smooth manifold); moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087540/s08754030.png" /> can be chosen odd.

====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 {{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>