[Math] When does homology represent an embedded sphere

Question

If we have a triangulation of a manifold $M$ of dimension $i$ and we have simplicial homology $H_i(M)=\mathbb{Z}$, what is the condition than there exists an embedded sphere $S^i$ that generates the homology group? We could also have that $H_i(M)$ is of higher rank, when do there exists spheres that generate a $\mathbb{Z}$ factor? Does this question even make sense and can the answer be simpler in case when $M$ is of low dimension, 2 or 3, or $i$ is low, 1 or 2?

Answer 1

(I'm assuming that the dimension $\dim(M)$ should have been different than the dimension of the homology class--otherwise this can only happen if $M$ is the disjoint union of $S^i$ with some other manifold.)

This is a very classical question. Here are some things you can say about it:

• In order for this to be possible, your element in $w \in H_i(M)$ needs to be in the image of the Hurewicz map $$\pi_i(M) \to H_i(M).$$ This is simply not always possible. For example, there is an exact sequence involving group homology: $$ \pi_2(M) \to H_2(M) \to H_2(\pi_1(M); \Bbb Z) \to 0 $$ This means that the image of $w$ in $H_2(\pi_1(M); \Bbb Z)$ needs to be zero.

• If $M$ has no homotopy groups below degree $i$, $w$ is always in the Hurewicz image: this is the Hurewicz theorem.

• One necessary condition on your homology class is that it must be primitive: the diagonal $\Delta: X \to X \times X$ takes $w$ to $i_*(w) + j(w)$, where $i$ and $j$ are inclusions $X \to X \times X$ as horizontal and vertical slices.

• After finding whether your homology class comes from homotopy, you need to check whether it comes from an embedding. If $i < \dim(M)/2$, I believe that this is always possible by a variant of the Whitney embedding theorem (you perturb your map to be a smooth embedding).

• It gets very difficult and situational if $i \geq \dim(M)/2$. For 3-dimensional manifolds, the sphere theorem allows one to find nonzero elements of $\pi_2(M)$ coming from embedded spheres, but this is a tough theorem.