Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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