## Osaka Journal of Mathematics

### Realizing homology classes up to cobordism

#### Abstract

It is known that neither immersions nor maps with a fixed finite set of multisingularities are enough to realize all mod $2$ homology classes in manifolds. In this paper we define the notion of realizing a homology class up to cobordism; it is shown that for realization in this weaker sense immersions are sufficient, but maps with a fixed finite set of multisingularities are still insufficient.

#### Article information

Source
Osaka J. Math., Volume 54, Number 4 (2017), 801-805.

Dates
First available in Project Euclid: 20 October 2017

https://projecteuclid.org/euclid.ojm/1508486581

Mathematical Reviews number (MathSciNet)
MR3715364

Zentralblatt MATH identifier
06821139

#### Citation

Grant, Mark; SZŰCS, András; Terpai, Tamás. Realizing homology classes up to cobordism. Osaka J. Math. 54 (2017), no. 4, 801--805. https://projecteuclid.org/euclid.ojm/1508486581

#### References

• V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko: Singularities of Differentiable Maps, Volume 1: Classification of Critical Points, Caustics and Wave Fronts, Birkhäuser, 1985.
• A. Clement: Integral Cohomology of Finite Postnikov Towers, Thèse de doctorat, Université de Lausanne, 2002.
• F.R. Cohen, T.J. Lada and J.P. May: The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer, 1976.
• P.E. Conner and E.E. Floyd: Differentiable periodic maps, Bull. Amer. Math. Soc. 68 (1962), 76–86.
• S. Eilenberg: On the problems in topology, Ann. of Math. (2) 50 (1949), 246–260.
• M. Grant and A. Sz\H ucs: On realising homology classes by maps of restricted complexity, Bull. London Math. Soc. 45 (2013), 329–340.
• J.P. May: The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer, 1972.
• J.W. Milnor and J.C. Moore: On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264.
• D.J. Pengelley and F. Williams: Global structure of the mod two symmetric algebra, $H^*(BO; \mathbb F_2)$, over the Steenrod Algebra, Alg. Geom. Topol. 3 (2003), 1119–1139.