Difference between revisions of "Steenrod problem"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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 |
Steenrod problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steenrod_problem&oldid=24570