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There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic.

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine.

There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere. But in general we may get many different boundaries depending on the choice of isotopy class and framing of the boundary of the cores of the 2-handles.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic. Added: See an explicit example due to Rylee Lyman in the comments. A simpler presentation of the Higman group is given (which perfect and has aspherical presentation complex).

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine. The only obvious restriction I see is that the fundamental group must have cohomological dimension three.

There are plenty of such manifolds, but as Danny indicates in his answer, there is likely not a classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic.

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine.

In the case of empty boundary, there is even less known about the classification, although examples exist. For example, I'm not sure if it's known that finitely presented $PD(4)$ groups are the fundamental groups of $4$-manifolds (see Question 3.4). Also, the Borel conjecture (that homotopy equivalent aspherical closed 4-manifolds are homeomorphic) is wide open in four dimensions.

There are plenty of such manifolds, but as Danny indicates in his answer, there is not a known classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic.

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine.

There are plenty of such manifolds, but as Danny indicates in his answer, there is likely not a classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic.

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine.

In the case of empty boundary, there is even less known about the classification, although examples exist. For example, I'm not sure if it's know that finitely presented $PD(4)$ groups are the fundamental groups of $4$-manifolds (see Question 3.4).

There are plenty of such manifolds, but as Danny indicates in his answer, there is likely not a classification.

Take any acyclic group $G$ with a finite aspherical 2-complex $C$ with $\pi_1(C)=G$. Then one can create an aspherical 4-manifold with boundary having $G$ as fundamental group. We may assume that the 1-skeleton $C^{(1)}$ of $C$ is a wedge of $k$ circles. Then take a 4-dimensional handlebody $H$ with $k$ 1-handles, with a spine of $C^{(1)}$. There are $k$ disks attached to the 1-skeleton in $C$. Attach $2$-handles to $H$ in such a way that the core of the attaching map is homotopic to the attaching map in the 2-skeleton $C^{(1)}$ to get a manifold $W$ with handle structure so that $C$ is a deformation retract of $W$, and hence $\pi_1(W)\cong G$. By the Poincaré-Lefschetz theorem, $\partial W$ is a homology 3-sphere.

To get such groups $G$, one can choose a small-cancellation $C'(\frac16)$ balanced presentation with $H_1(G)=0$. Then a presentation complex $C$ will be aspherical and acyclic.

The difficulty here is that one has no idea what 3-manifold the boundary of such a manifold will be. Moreover, it's not clear what the homeomorphism classification of such manifolds is, even if they have the same aspherical 2-skeleton spine and boundary.

Presumably there are also examples which do not have a 2-dimensional spine.

In the case of empty boundary, there is even less known about the classification, although examples exist. For example, I'm not sure if it's known that finitely presented $PD(4)$ groups are the fundamental groups of $4$-manifolds (see Question 3.4). Also, the Borel conjecture (that homotopy equivalent aspherical closed 4-manifolds are homeomorphic) is wide open in four dimensions.