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Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is in the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).

Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is in the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).

Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).

Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is in the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).

Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X_{g'}$, $X_g \cup_{\psi} X_{g'}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X_{g'}$, $X_g \cup_{\psi} X_{g'}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X_{g'} \rightarrow X_g \cup_{\psi} X_{g'}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X_{g'} \rightarrow X_g \cup_{\psi} X_{g'}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X_{g'} \rightarrow X_g \cup_{\psi} X_{g'}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).

Yes I did mean the article by Birman :) It seems that are a few different types of equivalence, namely strong equivalence, equivalence, and weak equivalence.

Let $X_g$ be the handlebody of genus g that is oriented, and $X'_g$ its homemorphic image, i.e. $X'_g = \tau(X_g)$. Let $\delta : \partial X_g \rightarrow \partial X_g$ be an arbitrary fixed orientation-reversing homeomorphism that extends to an orientation-reversing homeomorphism of the entire handle-body, $X_g \rightarrow X_g$, with the boundaries of the pair of handle-bodies identified by $\tau \delta \phi(p)=p$, for all elements, p of the handle-body $X_g$.

If two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are homeomorphic, then this is denoted by $\phi \equiv \psi$, with $\phi$ and $\psi$ being in the same isotopy class.

Two Heegaard splittings $X_g \cup_{\phi} X'_{g}$, $X_g \cup_{\psi} X'_{g}$ are called strongly equivalent (denoted by $\phi \approxeq \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that $h(X_g)=X_g$ and $h(X'_g)= X'_g$.

Heegaard splittings are called equivalent ($\phi \approx \psi$) if there is an orientation-preserving homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

Heegaard splittings are called weakly equivalent ($\phi$ $\sim$ $\psi$) if there is a homeomorphism $h:X_g \cup_{\phi} X'_{g} \rightarrow X_g \cup_{\psi} X'_{g}$ such that either $h(X_g)=X_g , h(X'_g)= X'_g$ ; or $h(X_g)=X'_g , h(X'_g)= X_g$.

And so $\phi \approxeq \psi \Rightarrow \phi \approx \psi \Rightarrow \phi \sim \psi \Rightarrow \phi \equiv \psi$

However, it is not the case that if there exists a homeomorphism between two Heegaard splittings that they are weakly equivalent, as there are examples of manifolds that have more than one weak equivalence class (which implies that homeomorphic Heegaard splittings need not be equivalent).

This is important as $\phi \approxeq \psi$ iff the isotopy classes of $\phi$ and $\psi$ respectively are in the same double coset of the group of isotopy classes of orientation-preserving self-homeomorphisms of $\partial X_g$ modulo the subgroup whose isotopy classes contain mappings that extend to homeomorphisms of $X_g$;

two Heegaard splittings are equivalent iff either the isotopy classes of $\phi$ and $\psi$ are in the same double-coset, or the isotopy class of $\psi$ is in the same double-coset as $\Delta \Phi \Delta ^{-1}$, where $\Delta$ is the isotopy class of $\delta$, the arbitrary fixed orientation-reversing homeomorphism, and $\Phi$ is the isotopy class of $\phi$;

and two Heegaard splittings are weakly equivalent iff the isotopy class of $\psi$ is the same double-coset as the isotopy class of $\phi$ ($\Phi$), or $\Delta \Phi ^{-1} \Delta ^{-1}$ or $\Phi^{-1}$ or $\Delta \Phi \Delta ^{-1}$ (For a proof of this, see "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds." in "Knots, groups, and 3-manifolds: papers dedicated to the memory of R. H. Fox" also by Joan Birman, from which this is a summary).

And so if there is a homeomorphism between two Heegaard splittings, that are not at least weakly equivalent, then the associated orientation-preserving homeomorphisms, $\psi$ and $\phi$ will not be in the same double-coset, as described above.

It seems difficult in general to determine whether two gluing maps will produce the same manifold, however there appear to be certain cases in which homological invariants can be used to show that particular manifolds have Heegaard splittings that are not homeomorphic (also in "On the equivalence of Heegaard splittings of closed, orientable 3-manifolds.").

There appear to be techniques to determine whether two Heegaard splittings are equivalent, which involve using the fundamental group of the handle-bodies, as there is a one-to-one correspondence between the equivalence classes of splitting homomorphisms and equivalence classes of Heegaard splittings; that is if the splitting homomorphisms induced by a pair of Heegaard splittings are equivalent, then the pair of Heegaard splittings are equivalent (See Jaco, William. Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc. 144 1969 365–379).