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Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold.

When starting to write this answer I found this related MO questionMO question. However, my question has to do with one of its answers (which is similar to the hint given in the Princeton Companion).

The proceedure for constructing this manifold from a given presentation is first to construct a CW-complex with that fundamental group (by wedge sum of circles puting 2-cells to cover the relations) embedding it in $R^5$ and considering the frontier of a tubular neighborhood.

Two questions come up to me (which are maybe trivial):

1- Why this cannot be done in $R^4$? Or can it be but the result is not the same?

2- Why the resulting manifold has the desired fundamental group?

Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold.

When starting to write this answer I found this related MO question. However, my question has to do with one of its answers (which is similar to the hint given in the Princeton Companion).

The proceedure for constructing this manifold from a given presentation is first to construct a CW-complex with that fundamental group (by wedge sum of circles puting 2-cells to cover the relations) embedding it in $R^5$ and considering the frontier of a tubular neighborhood.

Two questions come up to me (which are maybe trivial):

1- Why this cannot be done in $R^4$? Or can it be but the result is not the same?

2- Why the resulting manifold has the desired fundamental group?

Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold.

When starting to write this answer I found this related MO question. However, my question has to do with one of its answers (which is similar to the hint given in the Princeton Companion).

The proceedure for constructing this manifold from a given presentation is first to construct a CW-complex with that fundamental group (by wedge sum of circles puting 2-cells to cover the relations) embedding it in $R^5$ and considering the frontier of a tubular neighborhood.

Two questions come up to me (which are maybe trivial):

1- Why this cannot be done in $R^4$? Or can it be but the result is not the same?

2- Why the resulting manifold has the desired fundamental group?

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rpotrie
rpotrie
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Constructing 4-manifolds with fundamental group with a given presentation.

Reading Princeton Companion I found out that every finitely presented group can be realized as the fundamental group of a 4-manifold.

When starting to write this answer I found this related MO question. However, my question has to do with one of its answers (which is similar to the hint given in the Princeton Companion).

The proceedure for constructing this manifold from a given presentation is first to construct a CW-complex with that fundamental group (by wedge sum of circles puting 2-cells to cover the relations) embedding it in $R^5$ and considering the frontier of a tubular neighborhood.

Two questions come up to me (which are maybe trivial):

1- Why this cannot be done in $R^4$? Or can it be but the result is not the same?

2- Why the resulting manifold has the desired fundamental group?