Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Embedding obstructions and $4$-dimensional thickenings of $2$-complexes

Author: Vyacheslav S. Krushkal
Journal: Proc. Amer. Math. Soc. 128 (2000), 3683-3691
MSC (1991): Primary 57M20, 57Q35, 55S30, 57M25
Published electronically: May 18, 2000
MathSciNet review: 1690995
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial $n$-complex into ${\mathbb{R}}^{2n}$ for $n\neq 2$, and it was recently shown to be incomplete for $n=2$. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of $2$-complexes in ${\mathbb{R}}^4$.

References [Enhancements On Off] (What's this?)

  • 1. J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), No. 4, 445-453. MR 85d:57002
  • 2. E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. MR 25:4537
  • 3. M. H. Freedman, Are the Borromean rings $(A,B)$-slice?, Topology Appl. 24 (1986), 143-145. MR 88g:57015b
  • 4. M. H. Freedman, V. S. Krushkal and P. Teichner, Van Kampen's Embedding Obstruction is Incomplete for $2$-Complexes in ${{\mathbb{R}}}^{4}$, Math. Res. Lett. 1 (1994), No. 2, 167-176. MR 95c:57005
  • 5. A. Haefliger, Plongements différentiables dans le domaine stable, Comment. Math. Helv. 37 (1962/63), 155-176. MR 28:625
  • 6. D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966), 431-449. MR 34:2010
  • 7. C. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271-283.
  • 8. H. Sachs, On spatial representations of finite graphs, Finite and infinite sets, Vol. I, II, Colloq. Math. Soc. Janos Bolyai, 37, North-Holland, Amsterdam-New York, 1984. MR 87c:05055
  • 9. J. Segal, A. Skopenkov, S. Spiez, Embeddings of polyhedra in $\mathbb{R}^m$ and the deleted product obstruction, Topology Appl. 85 (1998), 335-344. MR 99b:57046
  • 10. A. Shapiro, Obstructions to the imbedding of a complex in a euclidean space I, Ann. of Math. 66 (1957) No.2, 256-269. MR 19:671a
  • 11. V. G. Turaev, Milnor invariants and Massey products, J. Soviet Math. 12 (1979), 128-137. MR 56:9538
  • 12. E. R. Van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Sem. Univ. Hamburg, vol.9 (1933), pp. 72-78 and 152-153.
  • 13. W. T. Wu, A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space, Science Press, Peking 1965.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57M20, 57Q35, 55S30, 57M25

Retrieve articles in all journals with MSC (1991): 57M20, 57Q35, 55S30, 57M25

Additional Information

Vyacheslav S. Krushkal
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520

Received by editor(s): May 20, 1998
Received by editor(s) in revised form: January 29, 1999
Published electronically: May 18, 2000
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society