Homology 3-spheres not embedding in the 4-sphere

 A natural question in topology is whether given two manifolds $N^n$ and $M^m$ with $m > n$, does $N$ embed in $M$.  There's an interesting history to this problem in a variety of dimensions, but I'll focus on the case of embeddings of homology 3-spheres into $S^4$ due to recent work of Clayton McDonald.  

In the topological category, Freedman proved that every homology 3-sphere embeds topologically into $S^4$.  Indeed, every homology sphere bounds a contractible topological 4-manifold.  Doubling this contractible 4-manifold produces a copy of $S^4$, since a simply-connected 4-manifold with the same homology as $S^4$ is homeomorphic to $S^4$ by Freedman's classification of closed, simply-connected, topological 4-manifolds.    

The smooth case is more interesting, as some homology 3-spheres do not embed smoothly in $S^4$, while some do.  From now on, we work only in the smooth category.  Of course $S^3$ embeds smoothly in $S^4$, while the Rokhlin invariant obstructs the Poincare homology sphere from embedding in $S^4$.  In fact, a homology sphere with non-trivial Rokhlin invariant cannot bound a smooth integer homology ball (since a homology ball is spin and has signature zero), and thus cannot embed in a homology sphere.  

Most invariants that obstruct homology spheres from embedding in $S^4$ (e.g. the Rokhlin invariant, Heegaard Floer d-invariant, Froyshov h-invariant, equivariant homology cobordism invariants of Manolescu / Lin / Hendricks-Manolescu) apply to obstruct homology spheres from embedding in homology 4-spheres.  However, recent work of McDonald (https://arxiv.org/abs/2202.02696) constructs homology 3-spheres which embed in homology 4-spheres, but not any homotopy 4-sphere.  First, I'll describe the *obstruction*, which comes Daemi's $\Gamma$-invariant (https://arxiv.org/abs/1810.08176), then the *construction*, which comes from symmetric unions of knots.  

Let $Y$ be a homology 3-sphere that bounds a definite 4-manifold with non-standard intersection form.  Taubes proves that for any definite 4-manifold $W$ with boundary $Y \# - Y$, there exists a non-trivial $SU(2)$-representation of $\pi_1(W)$.  More generally, Daemi proves that if $Y$ has non-vanishing Froyshov invariant (e.g. if $Y$ bounds non-standard, or also conjecturally $d(Y) \neq 0$), then some non-trivial representation of $\pi_1(Y \# - Y)$ extends over $W$.  For the application at hand, we take $Y$ to be the Poincare homology sphere $P$.    

The key construction is the symmetric union.  Given the standard ribbon disk for $K \# -K$, we can add twists to the ribbon bands to produce a new knot called a symmetric union of $K$.  If the twists in the band are full twists, then the symmetric union $J$ (called an even symmetric union) is a slice of the spin of $K$, a particular 2-knot in $S^4$.  Furthermore, $\Sigma_2(J)$, the 2-fold branched cover of $J$, embeds in the spin of $\Sigma_2(K)$; if $\Sigma_2(K)$ is a homology 3-sphere, then the spin of $\Sigma_2(K)$ will be a homology 4-sphere.  Finally, if the twists in the even symmetric union $J$ are all of the same sign, then there is a definite 2-handle cobordism $W$ from $\Sigma_2(K \# -K)$ to $\Sigma_2(J)$.  If $\Sigma_2(K)$ happens to be a homology sphere, then the same is true of $\Sigma_2(J)$.  For symmetric unions with "enough twists", the fundamental group of $W$ is in fact given by $\pi_1(\Sigma_2(K))$.  

Now we take $K = T_{3,5}$, so $\Sigma_2(K) = P$.  Hence, any even symmetric union $J$ obtained from $K$ has the property that its 2-fold branched cover embeds in a homology 4-sphere.  If the twists all have the same sign (and there "enough twists"), then we can construct a definite cobordism $W$ from $\Sigma_2(K \# - K) = P \# - P$ to $\Sigma_2(J)$ with fundamental group given by $\pi_1(P)$, which is the binary icosahedral group.  The proof is then completed as follows.  If $\Sigma_2(J)$ were to embed in a homotopy 4-sphere, then one half of this homotopy sphere could be glued onto $W$ to produce a definite manifold with boundary $P \# - P$ and the fundamental group admits no non-trivial $SU(2)$ representations.  More precisely, if $X_1$ and $X_2$ are the two halves of the homotopy sphere that $\Sigma_2(J)$ embeds in, then each of $X_i \cup W$ admits $SU(2)$ representations extending one of $P \# - P$; a Seifert-van Kampen argument applied to $X_1 \cup X_2 \cup W$ then shows that for $X_1 \cup X_2$ to be a homotopy sphere, the image of $\pi_1(W)$ in $\pi_1(X_i \cup W)$ must be a proper quotient of $\pi_1(W)$ for some $i$.  However, the only non-trivial quotient of the binary icosahedral group is $A_5$ which admits no non-trivial $SU(2)$ representations.  Hence, for this $i$, $\pi_1(X_i \cup W)$ does not admit a non-trivial $SU(2)$ representation extending one on the boundary.  This violates Daemi's theorem.  Hence, for the suitable symmetric union $\Sigma_2(J)$ embeds in a homology 4-sphere but not a homotopy 4-sphere.

Since we do not know any examples of homotopy 4-spheres which are not diffeomorphic to $S^4$, it would be interesting to see an obstruction that uses something more special to $S^4$ than being a homotopy sphere.