Oriented cobordism classes represented by rational homology spheres

The necessary condition pointed out by Jens Reinhold is also sufficient: any torsion class $x = [M] \in \Omega^{SO}_d$ admits a representative where $M$ is a rational homology sphere.

EDIT: This is Theorem 8.3 in $\Lambda$-spheres by Barge, Lannes, Latour, and Vogel. They also calculate the group of rational homology spheres up to rational h-cobordism, and more. I'll leave my argument below:

To prove this, we first dispense with low-dimensional cases: in any dimension $d < 5$ the only torsion class is $0 = [S^d]$. The high dimensional case follows from Claims 1 and 2 below.

I'll write $MX$ for the Thom spectrum of a map $X \to BO$ and $\Omega^X_d \cong \pi_d(MX)$ for the bordism group of smooth $d$-manifolds equipped with $X$-structure. Representatives are smooth closed $d$-manifolds $M$ with some extra structure, which includes a continuous map $f: M \to X$.

Claim 1: if $d \geq 5$ and $X$ is simply connected and rationally $\lfloor d/2 \rfloor$-connected, then any class in $\Omega^X_d$ admits a representative where $M$ is a rational homology sphere.

Claim 2: There exists a simply connected space $X$ such that $\widetilde{H}_*(X;\mathbb{Z}[\frac12]) = 0$, and map $X \to BSO$ such that the image of the induced map $\Omega^X_d = \pi_d(MX) \to \pi_d(MSO) = \Omega_d^{SO}$ is precisely the torsion subgroup, for $d > 0$.

Proof of Claim 1: Starting from an arbitrary class in $\Omega^X_d$ we can use surgery to improve the representative. Since $X$ is simply connected and $d > 3$ we can use connected sum and then surgery on embeddings $S^1 \times D^{d-1} \hookrightarrow M$ to make $M$ simply connected. Slightly better, such surgeries can be used to make the map $M \to X$ be 2-connected, meaning that its homotopy fibers are simply connected. From now on we need not worry about basepoints and will write $\pi_{k+1}(X,M) = \pi_k(\mathrm{hofib}(M \to X))$. These are abelian groups for all $k$.

If there exists a $k < \lfloor d/2\rfloor$ with $\widetilde{H}_k(M;\mathbb{Q}) \neq 0$ we can choose $\lambda \in H_k(M;\mathbb{Q})$ and $\mu \in H_{d-k}(M;\mathbb{Q})$ with intersection number $\lambda \cdot \mu \neq 0$. If $d = 2k$ for even $k$ we can additionally assume $\lambda \cdot \lambda = 0$, since the signature of $M$ vanishes. The rational Hurewicz theorem implies that $\pi_k(M) \otimes \mathbb{Q} \to H_k(M;\mathbb{Q})$ is an isomorphism, and the long exact sequence implies that $\pi_{k+1}(X,M) \otimes \mathbb{Q} \to \pi_k(M)\otimes\mathbb{Q}$ is surjective. After replacing $\lambda$ by a non-zero multiple, we may therefore assume that it admits a lift to $\pi_{k+1}(X,M)$. Such an element can be represented by an embedding $j: S^k \times D^{d-k} \hookrightarrow M$, together with a null homotopy of the composition of $j$ with $M \to X$. In the case $k < d/2$ this follows from Smale-Hirsh theory, in the case $d = 2k$ we must also use $\lambda \cdot \lambda = 0$ to cancel any self-intersections. (Actually there could also be obstructions to this in the case $d=2k$ for odd $k$, but those obstructions vanish after multiplying $\lambda$ by 2.) The embedding and the nullhomotopy gives the necessary data to perform surgery on $M$ and to promote the surgered manifold to a representative for the same class in $\Omega^X_d$.

Performing the surgery gives a new manifold $M'$ where $H_k(M';\mathbb{Q})$ has strictly smaller dimension than $H_k(M;\mathbb{Q})$ and $\widetilde{H}_*(M';\mathbb{Q}) = 0$ for $* < k$. This is seen in the same way as in Kervaire-Milnor. The case $d > 2k+1$ is easy, similar to their Lemma 5.2. In the case $d = 2k+1$ the diagram on page 515 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_k(M')$. In the case $d = 2k$ the diagram on page 527 shows that we can kill the homology class $j[S^k]$ and at worst create some new torsion in $H_{k-1}(M')$.

In finitely many steps we arrive at a representative where $\widetilde{H}_k(M;\mathbb{Q}) = 0$ for all $k \leq \lfloor d/2\rfloor$. Poincaré duality then implies that $H_*(M;\mathbb{Q}) \cong H_*(S^d;\mathbb{Q})$. $\Box$.

Proof of Claim 2: Finiteness of the stable homotopy groups of spheres implies that $\pi_d(MX)$ is a torsion group for $d > 0$ for any such $X$. Therefore we can never hit more than the torsion in $\pi_d(MSO)$, all of which is exponent 2 by Wall's theorem. The difficult part is to construct an $X$ where all torsion is hit.

The non-trivial based map $S^1 \to BO$ factors through $\mathbb{R} P^\infty \to BO$, whose image in mod 2 homology generates the Pontryagin ring $H_*(BO;\mathbb{F}_2)$. We can freely extend to double loop maps $$\Omega^2 S^3 \to \Omega^2 \Sigma^2 \mathbb{R}P^\infty \to BO$$ where the second map then induces a surjection on mod 2 homology. Both $\Omega^2 \Sigma^2 \mathbb{R}P^\infty$ and $BO$ split as $\mathbb{R} P^\infty$ times their 1-connected cover, so the induced map of 1-connected covers $\tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty) \to \tau_{\geq 2}(BO) = BSO$ also induces a surjection on mod 2 homology.

Now let $X = \tau_{\geq 2}(\Omega^2 \Sigma^2 \mathbb{R}P^\infty)$ with the map to $BSO$ constructed above. Take 1-connected covers of the double loop maps above, Thomify, 2-localize, and use the Hopkins-Mahowald theorem to get maps of $E_2$ ring spectra $$H \mathbb{Z} _{(2)} \to MX_{(2)} \to MSO_{(2)}.$$ (See e.g. section 3 of this paper.) We can view $MX_{(2)} \to MSO_{(2)}$ as a map of $H\mathbb{Z}_{(2)}$-module spectra, and hence $MX/2 \to MSO/2$ as a map of $H\mathbb{F}_2$-module spectra. The induced map $H_*(MX/2;\mathbb{F}_2) \to H_*(MSO/2;\mathbb{F}_2)$ is still surjective (it looks like two copies of $H_*(X;\mathbb{F}_2) \to H_*(BSO;\mathbb{F}_2))$, and inherits the structure of a module map over the mod 2 dual Steenrod algebra $\mathcal{A}^\vee = H_*(H\mathbb{F}_2;\mathbb{F}_2)$. Both modules are free, because any $H\mathbb{F}_2$-module spectrum splits as a wedge of suspensions of $H\mathbb{F}_2$. In fact the Hurewicz homomorphism $\pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2)$ induces an isomorphism $$\mathcal{A}^\vee \otimes \pi_*(MX/2) \to H_*(MX/2;\mathbb{F}_2),$$ and similarly for $MSO$. Therefore the map $\pi_*(MX/2) \to \pi_*(MSO/2)$ may be identified with the map obtained by applying $\mathbb{F}_2 \otimes_{\mathcal{A}^\vee} (-)$ to the map on homology, showing that the induced map $\pi_*(MX/2) \to \pi_*(MSO/2)$ is also surjective. Now any 2-torsion class $x \in \pi_d(MSO)$ comes from $\pi_{d+1}(MSO/2)$, hence from $\pi_{d+1}(MX/2)$ and in particular from $\pi_d(MX)$. $\Box$

Edit: I realize now that the surgeries may have not preserved the finiteness of the homotopy groups I wish to be finite. This needs to be addressed in order for this to be correct.

Suppose $M$ is a $2k+1$ dimensional manifold with the assumption that $\pi_n (M)$ is finite for $n=0,4$ modulo 8 if $n \leq k$. In low dimensions (particularly less than 8) this is very easy to satisfy (and check). Also, the fundamental group of $M$ should be abelian. We will show $M$ is cobordant to a rational homology sphere. Of course, the idea of surgery is to kill homology groups, and we would like to kill the free part of the homology of $M$ if possible in order to obtain the result. However, with no normal invariant information it is impossible to guarantee the free part of the homology is even spherical.

So the idea is instead we should focus on the homotopy groups, do surgery to make them torsion below the middle dimension, and then apply mod C theory to conclude that the homology groups are also finite (this is why we require the fundamental group abelian).

Since we are below the middle dimension, every class we want to do surgery on is represented by an embedded sphere. And for dimensional reasons we can work purely with stable normal bundles.

Start with an embedding $f \in I_n (M)$ where $I_n(M)$ denotes the regular homotopy classes of framed immersions with addition connect sum, and suppose $n<k$. We should have $f$ nontorsion in the homotopy groups, since otherwise there is no reason to do surgery on it. The normal bundle need not be trivial, but we can of course consider multiples $hf$ of $f$ in $I_n(M)$. By proposition 5.55 in Ranicki's Algebraic and Geometric Surgery, the normal bundle of $hf$ is $h$ times the normal bundle of $f$ viewed as a map into $BO$.

By our assumptions, there is some $h$ so that the normal bundle of $hf$ is trivial. This follows from the computations of the homotopy groups of $BO$, they are not torsion only in dimensions $2,6$. Now do surgery on $hf$ with the effect of quotienting by the smallest subgroup containing $hf$ closed under the action of the fundamental group. So necessarily, we have reduced the rank of our group. Repeating in this manner, we can make $M$ have strictly torsion homotopy groups below dimension $k$.

In dimension $k$, it is slightly trickier since surgeries can create new elements in the homotopy group. However, we can proceed almost exactly as Kervaire and Milnor did, around lemma 5.6 in "Groups of Homotopy Spheres".

By mod C theory, the rank of the kth homology and the rank of the kth homotopy of $M$ (of course this is a surgered $M$) are the same. In particular, this is seen through the Hurewicz map. We have that any nontorsion element $x$ in the homology has some multiple $hx$ that is hit by the Hurewicz map and of course this can be represented as an embedding. From lemma 5.6 we can deduce that $H_k(M)/hx$ is isomorphic to $H_k(M')/T$ where $T$ is some torsion group, where $M'$ is the result of the surgery on $hx$. This implies that the rank of $H_k(M')$ is smaller than $H_k(M)$ since $x$ was nontorsion.

Proceeding in this manner, we can obtain a torsion homology group, and so also a torsion homotopy group. Of course, we then apply mod C theory to deduce all the homology below the middle dimension is torsion, and so by Poincare duality all homology is torsion. Hence, $M$ can be surgered to be a rational homology sphere.

Under the same conditions, I think without too much trouble we can extend to even dimensions. Though I do not think we can work purely with stable normal bundle when we get to the middle dimension. One would like to also convert the conditions on $\pi_n(M)$ modulo 8 to a complete obstruction to being cobordant to a rational homology sphere.

Edit: After correcting the dimension restriction on the homotopy groups, it appears one can allow for torsion free elements in these groups if the Pontryagin class in the same degree in cohomology vanishes. This should follow from the Pontryagin classes detecting the nontorsion in $BO$.