Representability of the sum of homology classes

Question

This is probably a very simple question, but I have not found it addressed in the references that I know. Let $M$ be a closed and connected orientable $d$-dimensional manifold ($d\leq 8$) and let $[\alpha_{i}]\in H_{k}(M,\mathbb{Z})$, $k\leq 6$, $i=1,2,$ be $k$-homology classes of $M$ that admit representations $\iota_{i}\colon Z_{i}\hookrightarrow M$ in terms of closed oriented submanifolds $Z_{i}$, that is:

$\iota_{i \ast}[Z_{i}] = [\alpha_{i}]$

Since $k\leq 6$, every element in $H_{k}(M,\mathbb{Z})$ is representable by a closed oriented submanifold, so in particular $[\alpha] := [\alpha_{1}] + [\alpha_{2}]\in H_{k}(M,\mathbb{Z})$ is also representable by a closed oriented submanifold $\iota\colon Z\hookrightarrow M$. Is there any relation between $Z$ and $Z_{i}$? Can I "construct" a representative of $[Z]$ once I know $Z_{1}$ and $Z_{2}$?

Thanks.

Answer 1

There are a few well-known (at least to low-dimensional topologists) cases of this.

First, the obvious observation that if $k<d/2$, then one may take $Z_1\cap Z_2=\emptyset$ by a small isotopy, since this is the transversal case, so $Z=Z_1\cup Z_2$.

For $d=2, k=1$, this is well-known: we may assume $Z_1 \pitchfork Z_2$, so $Z_1\cap Z_2$ is a finite collection of points with transverse intersections modeled on $\{(x,y) \in \mathbb{R}^2 | xy=0\}$. We resolve the crossings by perturbing to $\{(x,y)\in \mathbb{R}^2 | xy = \epsilon\}$. The choice of the sign of $\epsilon$ depends on the local orientations at the points of $Z_1\cap Z_2$.

For $d=3, k=2$, this is known as double-curve sum to 3-manifold topologists. If $Z_1\pitchfork Z_2$, then $Z_1\cap Z_2$ is a collection of curves, locally modeled on $\{(x,y,z)\in \mathbb{R}^3 | xy=0\}$ (so the previous example crossed with $\mathbb{R}$). The same resolution of intersections works: $\{(x,y,z) \in \mathbb{R}^3 | xy=\epsilon\}$. (in the picture, $Z_1=R, Z_2=S$.)

I believe that this pattern persists for codimension-one homology classes in all dimensions: when $k=d-1$ and $Z_1\pitchfork Z_2$, then $Z_1\cap Z_2$ should be locally modeled on $(Z_1\cap Z_2) \times \{xy=0\}$, and the ``double curve sum" should be $(Z_1\cap Z_2) \times \{xy=\epsilon\}$.

For $d=4, k=2$, when $Z_1\pitchfork Z_2$, then $Z_1\cap Z_2$ is finite, and locally modeled on $\{(x,y)\in \mathbb{C}^2 | xy=0\}$. Now the local orientations on $M, Z_1, Z_2$ are determined by the complex structure. Then the resolution is $\{(x,y)\in \mathbb{C}^2 | xy=\epsilon\}$ again. I don't know whether this resolution always works in codimension 2 for representable homology classes in general (crossed with $Z_1\cap Z_2$) - there may be a framing issue if the normal bundle of $Z_1\cap Z_2$ is non-trivial. I think this ought to also work in $d=8, k=4$, replacing $\mathbb{C}$ with the quaternions $\mathbb{H}$.

Answer 2

My paper is now available at http://arxiv.org/abs/1705.03836. It describes a construction for the codimension-1 (mod 2 coefficients) and codimension-2 (twisted integer coefficients) case.