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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexa stub entry, for the moment just to record some references <a href="https://ncatlab.org/nlab/revision/link+cobordism/1">v1</a>, <a href="https://ncatlab.org/nlab/show/link+cobordism">current</a>

   a stub entry, for the moment just to record some references

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2024
   • (edited Jul 24th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have a **question**: Doesn't the Pontrjagin theorem (or explicity the move [here](https://ncatlab.org/nlab/show/group-completed+configuration+space+of+points#FramedLinksCobordantToFramedUnknots)) show that all links ($S^1 \sqcup \cdots \sqcup S^1 \hookrightarrow \mathbb{R}^3$) are cobordant to an unlink (in fact: to an unknot)? MathOverflow seems to agree ([here](https://math.stackexchange.com/q/426482/58526)). But authors on link cobordism speak as if the cobordism classes of links are not easily characterized and some seem to contradict the above statement (e.g. *[On links not cobordant to split links](https://doi.org/10.1016/0040-9383(80)90017-8)*). I must be missing something.

   I have a question:

   Doesn’t the Pontrjagin theorem (or explicity the move here) show that all links ($S^1 \sqcup \cdots \sqcup S^1 \hookrightarrow \mathbb{R}^3$) are cobordant to an unlink (in fact: to an unknot)?

   MathOverflow seems to agree (here). But authors on link cobordism speak as if the cobordism classes of links are not easily characterized and some seem to contradict the above statement (e.g. On links not cobordant to split links). I must be missing something.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave sent this question to MathOverflow: [MO:q/475666](https://mathoverflow.net/q/475666/381)

   have sent this question to MathOverflow: MO:q/475666

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 24th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am suspecting now the issue is that early authors actually mean *link concordance* when saying "link cobordism", which is of course much more restrictive. I have added this and further comments to the entry. <a href="https://ncatlab.org/nlab/revision/diff/link+cobordism/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/link+cobordism/3">v3</a>, <a href="https://ncatlab.org/nlab/show/link+cobordism">current</a>

   I am suspecting now the issue is that early authors actually mean link concordance when saying “link cobordism”, which is of course much more restrictive. I have added this and further comments to the entry.

   diff, v3, current