
TL;DR
This paper constructs a specific embedding of a circle in five-dimensional space demonstrating that no 3-sphere can be topologically linked with it, revealing new insights into high-dimensional linking phenomena.
Contribution
It provides an explicit example of a topological embedding in -dimensional space where the third homotopy group of the complement vanishes, showing no linking with 3-spheres.
Findings
Existence of a circle embedding in -space with trivial -homotopy of the complement
No 3-sphere can be linked with this embedding
Advances understanding of high-dimensional linking and embedding properties
Abstract
There is a topological embedding such that . Therefore, no -sphere can be linked with .
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Linking topological spheres
Piotr Hajłasz
Piotr Hajłasz,
Department of Mathematics, University of Pittsburgh,
301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Abstract.
There is a topological embedding such that . Therefore, no -sphere can be linked with .
Key words and phrases:
Linking number, integer homology spheres, Cannon-Edwards theorem
2010 Mathematics Subject Classification:
Primary: 57Q45; Secondary: 57M30
P.H. was supported by NSF grant DMS-1800457.
Dedicated to MathOverflow
If is smoothly embedded in , , then there is an embedding of so that the two spheres are linked with the liking number equal . Now, if is an arbitrary topological embedding, then by the Alexander duality , so the homology of the complement is the same for all embeddings. Therefore, one might expect that is linked with some ()-dimensional sphere embedded into the complement of , so that this embedding of gives a generator of . This is the case when . Indeed, then and the existence of a linked sphere follows from the fact that that is the abelianization of , so a generator of is given by a mapping from a circle; this mapping can be approximated by an embedding leading to an embedded that is linked with with the linking number . However, the case when is more subtle.
The purpose of this short note is to construct a somewhat surprising example: there is a topological embedding such that for every embedding , the spheres and are unlinked. More precisely so the embedding of in the complement of is always homotopic to a constant map.
The result is a simple consequence of the celebrated Cannon-Edwards theorem [3, 4, 5] according to which the double suspension of the integer homology sphere is a topological sphere. This deep and counterintuitive result is often used to construct various counterexamples that are similar in the spirit to the one presented below.
Since the linking number of highly non-smooth topological spheres has recently been used in geometric analysis (cf. [6, 8, 7, 12]), the author hopes that this example will be of some interest.
Using a one point compactification of it suffices to prove the following result.
Theorem 1**.**
There is a topological embedding such that .
Proof.
It is well known that there are -dimensional integer homology spheres whose universal cover is . For example, there are particular constructions in [1, 2, 9] of hyperbolic integer homology spheres. Note that the universal cover of a hyperbolic -manifold is the hyperbolic space that is homeomorphic to . Other examples are listed in [10]. Let be such an integer homology sphere. Since the universal cover of is contractible, . According to the celebrated theorem of Cannon and Edwards [3, 4, 5], the double suspension of an integer homology sphere is homeomorphic to a topological sphere. Let be such a homeomorphism. is a deformation retract of the complement of the vertices of the suspension . Therefore, is also a deformation retract of the complement of the suspension of the vertices in . Denote the suspension of the vertices by , so is a deformation retract of and hence . is homeomorphic to . If is a homeomorphism, then is a topological embedding and clearly . The proof is complete. ∎
Remark 2**.**
Taking the higher suspensions of the hyperbolic integer homology sphere we obtain embeddings of , , such that , so no -sphere can be linked with .
Remark 3**.**
If in the above construction we replace the hyperbolic homology sphere by the standard Poincaré homology sphere , the linking number of the embedding of to will be the multiplicity of . Indeed, and hence the degree of a map is a multiple of .
Acknowledgements. The author would like to express his deepest gratitude to the Mathoverflow community; without their help this short note would have never been written. In particular he would like to thank Ian Agol, Jason DeBlois, Neil Hoffman, John Klein, Thilo Kuessner, Andrew Putman, Daniel Ruberman, see [10, 11]. The author is also grateful to the referee for the valuable comments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 6[6] Goldstein, P., Hajłasz, P.: Jacobians of W 1 , p superscript 𝑊 1 𝑝 W^{1,p} homeomorphisms, case p = [ n / 2 ] 𝑝 delimited-[] 𝑛 2 p=[n/2] . (2018) ar Xiv:1812.11888
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- 8[8] Hajłasz, P., Mirra, J., Schikorra, A.: Hölder continuous mappings, differential forms and the Heisenberg groups. (In preparation.)
