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• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to one of the answers.

• The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. [The proof described there applies also to topological manifolds.]

[Further aside: Incidentally, the converse also holds — although we will not make use of it. In another mathoverflow question, Igor Belegradek remarks in his answer and the ensuing comments that if $M\times\RR$ is homeomorphic to $N\times\RR$, then $M$ and $N$ are $h$-cobordant.]

• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to one of the answers.

• The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. [The proof described there applies also to topological manifolds.]

[Further aside: Incidentally, the converse also holds — although we will not make use of it. In another mathoverflow question, Igor Belegradek remarks in his answer and the ensuing comments that if $M\times\RR$ is homeomorphic to $N\times\RR$, then $M$ and $N$ are $h$-cobordant.]

[Edit: I have added some details, and a more explicit example by Milnor.]

We will make fundamental use of the following fact: if $M$ and $N$ are two $h$-cobordant, closed, smooth (respectively, topological) manifolds of dimension greater than 3, then $M\times\RR$ is diffeomorphic (respectively, homeomorphic) to $N\times\RR$. Here are a couple of quick references, both making use of the $s$-cobordism theorem:

• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to the answer.

• The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. The proof described there applies also to topological manifolds.

[Aside: Incidentally, the converse also holds — although we will not make use of it. In another mathoverflow question, Igor Belegradek remarks in his answer and the ensuing comments that if $M\times\RR$ is homeomorphic to $N\times\RR$, then $M$ and $N$ are $h$-cobordant.]

So it suffices to find closed manifolds $M$ and $N$ which are $h$-cobordant, yet not homeomorphic. As required by the question, we also want $M$ to be null-cobordant; then $N$ will be null-cobordant as well. As a minor bonus, in all the examples presented below, $M$ and $N$ are smooth manifolds which are smoothly $h$-cobordant, so $M\times\RR$ is actually diffeomorphic to $N\times\RR$.

One such example is given by Farrel and Hsiang in their article "$H$-cobordant manifolds are not necessarily homeomorphic". There, the manifold $M$ is a product $M=L\times\TT^n$, where $L$ is any 3-dimensional lens space such that $\pi_1(L)\simeq\ZZ/p^2$ for some prime $p$, and $\TT^n=(S^1)^{\times n}$ is a torus of dimension $n\geq 3$. Note that $M$ is null-cobordant since the torus $\TT^n$ is null-cobordant. Unfortunately, Farrel and Hsiang do not explicitly describe $N$.

I will briefly describe how to conclude from Milnor's article that $M$ and $N$ above actually meet our requirements. In section 2, Milnor sketches a proof that $M$ and $N$ are $h$-cobordant. At the end of section 4, in corollary 2, he uses the fact that $L(7,1)$ and $L(7,2)$ have distinct Reidemeister torsion to conclude that $M$ and $N$ have distinct Reidemeister torsion themselves, and thus are not diffeomorphic. But more is true: since the Reidemeister torsions of $M$ and $N$ differ, $M$ and $N$ are not homeomorphic. This follows from the topological invariance (i.e. invariance under homeomorphisms) of Reidemeister torsion, which is itself a simple consequence of the topological invariance of Whitehead torsion — proved by Chapman after the publication of Milnor's article.

[Edit: I have added some details and a more explicit example by Milnor.]

We will make fundamental use of the following fact: if $M$ and $N$ are $h$-cobordant, closed, smooth manifolds of dimension greater than 3, then $M\times\RR$ is diffeomorphic to $N\times\RR$. [Aside: A similar statement holds for topological manifolds; simply replace diffeomorphism with homeomorphism.] Here are a couple of quick references for this fact, both making use of the $s$-cobordism theorem:

• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to one of the answers.

• The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. [The proof described there applies also to topological manifolds.]

[Further aside: Incidentally, the converse also holds — although we will not make use of it. In another mathoverflow question, Igor Belegradek remarks in his answer and the ensuing comments that if $M\times\RR$ is homeomorphic to $N\times\RR$, then $M$ and $N$ are $h$-cobordant.]

So it suffices to find closed smooth manifolds $M$ and $N$ which are $h$-cobordant, yet not homeomorphic. As required by the question, we also want $M$ to be null-cobordant; then $N$ will be null-cobordant as well. As a minor bonus, since $M$ and $N$ are smoothly $h$-cobordant, $M\times\RR$ is actually diffeomorphic to $N\times\RR$.

One such example is given by Farrel and Hsiang in their article "$H$-cobordant manifolds are not necessarily homeomorphic". There, the manifold $M$ is a product $M=L\times\TT^n$, where $L$ is any 3-dimensional lens space such that $\pi_1(L)\simeq\ZZ/p^2$ for some prime $p$, and $\TT^n=(S^1)^{\times n}$ is a torus of dimension $n\geq 3$. Note that $M$ is null-cobordant since the torus $\TT^n$ is null-cobordant. Unfortunately, the methods used by Farrel and Hsiang do not yield an explicit description of the manifold $N$.

I will briefly describe how to conclude from Milnor's article that the preceding $M$ and $N$ meet our requirements. In section 2, Milnor sketches a proof that $M$ and $N$ are $h$-cobordant. At the end of section 4, in the proof of corollary 2, he uses the fact that $L(7,1)$ and $L(7,2)$ have distinct Reidemeister torsions, and concludes that $M$ and $N$ also have distinct Reidemeister torsions, and thus are not diffeomorphic. But more is true: since the Reidemeister torsions of $M$ and $N$ differ, $M$ and $N$ are not homeomorphic. This follows from the topological invariance (i.e. invariance under homeomorphisms) of Reidemeister torsion, which is itself a simple consequence of the topological invariance of Whitehead torsion — proved by Chapman after the publication of Milnor's article.

We will make fundamental use of the following fact: if $M$ and $N$ are two $h$-cobordant closed topological manifolds of dimension greater than 3, then $M\times\RR$ is homeomorphic to $N\times\RR$. Here are a couple of quick references, both making use of the $s$-cobordism theorem:

• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to the answer.

• The article "Contact manifolds with symplectomorphic symplectizations" states such a result for smooth manifolds as corollary 2.5. Nevertheless, the proof described there applies also to topological manifolds.

So it suffices to find closed manifolds $M$ and $N$ which are $h$-cobordant, yet not homeomorphic. As required by the question, we also want $M$ to be null-cobordant; then $N$ will be null-cobordant as well.

One such example is given by Farrel and Hsiang in their article "$H$-cobordant manifolds are not necessarily homeomorphic". There, the manifold $M$ is a product $M=L\times\TT^n$, where $L$ is any $3$-dimensional lens space such that $\pi_1(L)\simeq\ZZ/p^2$ for some prime $p$, and $\TT^n=(S^1)^{\times n}$ is a torus of dimension $n\geq 3$. Note that $M$ is null-cobordant since the torus $\TT^n$ is null-cobordant. Unfortunately, Farrel and Hsiang do not explicitly describe $N$.

For instance, we can take $K$ and $L$ to be $3$-dimensional lens spaces, and not just $L(7,1)$ and $L(7,2)$. The homotopical classification and Reidemeister torsions of lens spaces are well-known: see http://www.map.mpim-bonn.mpg.de/Lens_spaces.

We will make fundamental use of the following fact: if $M$ and $N$ are two $h$-cobordant, closed, smooth (respectively, topological) manifolds of dimension greater than 3, then $M\times\RR$ is diffeomorphic (respectively, homeomorphic) to $N\times\RR$. Here are a couple of quick references, both making use of the $s$-cobordism theorem:

• In a related mathoverflow question, Oscar Randal-Williams provides a fairly direct argument in a comment to the answer.

• The article "Contact manifolds with symplectomorphic symplectizations" states the result for smooth manifolds as corollary 2.5. The proof described there applies also to topological manifolds.

So it suffices to find closed manifolds $M$ and $N$ which are $h$-cobordant, yet not homeomorphic. As required by the question, we also want $M$ to be null-cobordant; then $N$ will be null-cobordant as well. As a minor bonus, in all the examples presented below, $M$ and $N$ are smooth manifolds which are smoothly $h$-cobordant, so $M\times\RR$ is actually diffeomorphic to $N\times\RR$.

One such example is given by Farrel and Hsiang in their article "$H$-cobordant manifolds are not necessarily homeomorphic". There, the manifold $M$ is a product $M=L\times\TT^n$, where $L$ is any 3-dimensional lens space such that $\pi_1(L)\simeq\ZZ/p^2$ for some prime $p$, and $\TT^n=(S^1)^{\times n}$ is a torus of dimension $n\geq 3$. Note that $M$ is null-cobordant since the torus $\TT^n$ is null-cobordant. Unfortunately, Farrel and Hsiang do not explicitly describe $N$.

For instance, we can take $K$ and $L$ to be 3-dimensional lens spaces, and not just $L(7,1)$ and $L(7,2)$. The homotopical classification and Reidemeister torsions of lens spaces are well-known: see http://www.map.mpim-bonn.mpg.de/Lens_spaces.