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Cohomology and fundamental classes
Hello,
I would like to know if all homological classes in a smooth manifold can be represented as immersed submanifolds, or examples where this is not true and possible obstructions.
Thanks!!
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2$\begingroup$ The short answer is no and this is already a duplicate of at least a couple of questions already here, for example : mathoverflow.net/questions/21171/… $\endgroup$– Somnath BasuCommented Mar 3, 2011 at 7:16
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3$\begingroup$ Steenrod wanted to know the same thing. Thom managed to answer this (and much much more!) in his 1954 thesis. $\endgroup$– Dylan WilsonCommented Mar 3, 2011 at 7:17
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1$\begingroup$ Well, the question of which homology classes can be represented by immersions of closed manifolds is actually a bit more subtle than that asked in the question linked above, and answered by Thom. I think there are strict inclusions: homology classes representable by embedded $\subset$ immersed $\subset$ singular manifolds. $\endgroup$– Mark GrantCommented Mar 3, 2011 at 9:57
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5$\begingroup$ I find this strange. The previous questions ask about representing homology classes by submanifolds. This question asks about representing homology classes by immersed submanifolds. These are different questions with (I believe) different answers. (Or are we assuming that the OP meant to ask about submanifolds?) $\endgroup$– Mark GrantCommented Mar 3, 2011 at 12:00
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1$\begingroup$ I was asking about immersed manifolds, sorry for repeating questions and thanks again for the answers. $\endgroup$– shurtadosCommented Mar 4, 2011 at 2:55
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1 Answer
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It might be better to split the question into 2 cases and 2 steps.
Step 1: Which homology classes in $X$ can be represented by continuous maps of closed smooth manifolds (ie, which classes are $f_*[M]$ where $f\colon M\to X$ is a map from a closed manifold with fundamental class $[M]\in H_*(M)$)? This is the so-called 'Steenrod representability problem'.
Case 1: In the unoriented case (if you are asking about homology with mod $2$ coefficients) the answer is all of them. See Thom's paper "Quelques propriétés globales des variétés différentiables", or the book "Differentiable periodic maps" by Conner and Floyd.
Case 2: In the oriented case, we are asking which integral homology classes are represented by maps from closed orientable manifolds. The answer is not all of them, but positive multiples of all of them. More precise statements can be found in the papers of Thom ("Sur un problème de Steenrod" and "Quelques propriétés...") and their reviews.
Step 2: Now you have to ask which unoriented and oriented bordism classes contain immersions. To the best of my knowledge, this part is not completely known. But here is a reference to start looking:
Li, Gui Song "On immersions in bordism classes", Math. Ann. 291 (1991), no. 2, 373–382.
Note that in the introduction he quotes a result of Szűcs to the effect that, for any given oriented bordism class of sufficiently high codimension in a manifold, some positive multiple of it contains an immersion.
Update: Sorry to revive this old post, but I just wanted to shamelessly plug an article which arose directly from this question. In http://arxiv.org/abs/1111.0249 (to appear in Bulletin of the LMS) András Szűcs and myself show that in any codimension $k\ge 2$ there exists a closed smooth manifold $N^n$ and a mod $2$ homology class of dimension $n-k$ which cannot be realized by an immersion of a closed manifold. The proof employs explicit obstructions to realizability, involving Bocksteins and Steenrod squares (see Theorem 1.2).
answered Mar 3, 2011 at 7:24