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I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?

Realizing homology classes in a manifold via embedded submanifolds, I'm aware of the work of Serre and Thom, which gives sufficient conditions for realization, in terms of a range of dimensions, depending on coefficient group.

For singular homology of a particular CW-complex, my impression is the best way to go about this problem in practice would be the Atiyah-Hirzebruch spectral sequence for the singular bordism theory -- comparing to the cellular homology of the space in question.

The Connor and Floyd text has some strong results for singular realization. In this case the theorems give sufficient conditions for homology classes to be realizable by singular manifolds, in terms of the structure of the homology of the manifold -- odd torsion usually being the problem.

Are there some modern references that describe more recent and complete results?

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    $\begingroup$ Are any of the slightly more recent references in Section 6 of Sullivan, Dennis René Thom's work on geometric homology and bordism. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 341–350 ams.org/journals/bull/2004-41-03/S0273-0979-04-01026-2 of use to you? $\endgroup$
    – John Rognes
    Commented May 14 at 19:01
  • $\begingroup$ @JohnRognes: thanks, that's a nice survey. $\endgroup$
    – Ryan Budney
    Commented May 15 at 22:26

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A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, some integer multiple of every homology class is realizable by a map.

There has been recent work on trying to bound the integer we need to multiply by in the integral case. See for example the preprint A lower bound in the problem of realization of cycles by Vasilii Rozhdestvenskii, to appear in Journal of Topology.

For realizability by embeddings, Thom says a lot. There are some nice examples of non-realizable homology classes due to Bohr, Hanke and Kotschick in

Bohr, Christian; Hanke, Bernhard; Kotschick, Dieter, Cycles, submanifolds, and structures on normal bundles, Manuscr. Math. 108, No. 4, 483–494 (2002). ZBL1009.57043.

Note however that that paper contains a mis-statement in Remark 2 at the bottom of page 485, where it is claimed that collapse at the $E^2$ page of the Atiyah–Hirzebruch spectral sequence for oriented bordism implies that all integral homology is realizable by "immersed submanifolds". All we can really conclude is that everything is realized by a map, as in Steenrod's problem.

The situation for realizability by immersions is somewhat more subtle. András Szűcs and I gave the first examples of mod 2 homology classes not realizable by immersions in

Grant, Mark; Szűcs, András, On realizing homology classes by maps of restricted complexity, Bull. Lond. Math. Soc. 45, No. 2, 329–340 (2013). ZBL1270.57068.

We also have some results about non-realizability by maps with some presecribed set of (multi-)singularities, in the above paper and in

Grant, Mark; Szűcs, András, Homologies are infinitely complex, Topol. Methods Nonlinear Anal. 45, No. 1, 55–61 (2015). ZBL1368.57013.

Zhenhua Liu asked here on MathOverflow the very interesting (in my view) question of whether there are integral homology classes realizable by immersions but not embeddings: Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

A lot of these questions can be approched using classical (but nevertheless difficult) obstruction theory.

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  • $\begingroup$ Thanks Mark. This looks helpful. $\endgroup$
    – Ryan Budney
    Commented May 16 at 16:48
  • $\begingroup$ Thank you Mark, I am asking here since I noticed you worked a lot on the difference between embedded, immersed Thom criteria $\endgroup$
    – Crash Bandicoot
    Commented Sep 10 at 3:06
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    $\begingroup$ @CrashBandicoot: Yes every class in $H_5(M^8;\mathbb{Z})$ is represented as an embedding, because for the dual class $x\in H^3(M^8;\mathbb{Z})$ the image $St^5_3(x)\in H^8(M^8;\mathbb{Z})\cong\mathbb{Z}$ is $3$-torsion, and all the higher obstructions vanish for dimension reasons. $\endgroup$
    – Mark Grant
    Commented Sep 10 at 11:47
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    $\begingroup$ @CrashBandicoot The Steenrod power $St^5_3$, aka $\beta P^1_3$, is definitely to do with embeddings and not just Steenrod realizability. It's not until Chapitre 3 that Thom tackles Steenrod's problem by passing to an associated manifold (regular negihbourhood of an embedding in some Euclidean space) and the duals Poincare duals of the Steenrod powers come into play. $\endgroup$
    – Mark Grant
    Commented Sep 10 at 14:08
  • $\begingroup$ In answer to your last comment, any $z\in H_5(M^n;\mathbb{Z})$ is Poincare dual to a class in $x\in H^{n-5}(M^n;\mathbb{Z})$. The primary obstruction to embeddability is $St^5_3(x)\in H^n(M^n;\mathbb{Z})$ which vanishes for the reason I gave above. The higher obstructions will live in cohomology groups in degrees larger than $n$. $\endgroup$
    – Mark Grant
    Commented Sep 10 at 14:11

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