Abstract
We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ2n and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of ϑ is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of a given n-polyhedron in ℝ2n+1. (3) A “blown-up” one-parameter version of ϑ is a universal type 1 invariant of singular knots, i.e., knots in ℝ3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (= Polyak-Viro) formula. (4) Settling a problem of Yashchenko in the metastable range, we find that every PL manifold N nonembeddable in a given ℝm, m ≥ \( \frac{{3(n + 1)}} {2} \), contains a subset X such that no map N → ℝm sends X and N \ X to disjoint sets. (5) We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “k-co-connected and locally k-co-connected” polyhedra, which we embed in ℝ2n−k for k < \( \frac{{n - 3}} {2} \), thus extending the Penrose-Whitehead-Zeeman theorem.
Similar content being viewed by others
References
P. M. Akhmetiev, “Pontryagin-Thom Construction for Approximation of Mappings by Embeddings,” Topol. Appl. 140, 133–149 (2004).
P. M. Akhmet’ev and P. J. Eccles, “The Relationship between Framed Bordism and Skew-Framed Bordism,” Bull. London Math. Soc. 39, 473–481 (2007).
K. Barnett and M. Farber, “Topology of Configuration Space of Two Particles on a Graph. I,” arXiv: 0903.2180.
D. R. Bausum, “Embeddings and Immersions of Manifolds in Euclidean Space,” Trans. Am. Math. Soc. 213, 263–303 (1975).
J. C. Becker, “A Relation between Equivariant and Non-equivariant Stable Cohomotopy,” Math. Z. 199, 331–356 (1988).
M. Bestvina, M. Kapovich, and B. Kleiner, “Van Kampen’s Embedding Obstruction for Discrete Groups,” Invent. Math. 150, 219–235 (2002).
T. Böhme, “On Spatial Representations of Graphs,” in Contemporary Methods in Graph Theory, Ed. by R. Bodendieck (B.I.-Wissenschaftsverlag, Mannheim, 1990), pp. 151–167.
M. V. Brahm, “Approximating Maps of 2-Manifolds with Zero-Dimensional Nondegeneracy Sets,” Topol. Appl. 45, 25–38 (1992).
G. E. Bredon, Sheaf Theory, 2nd ed. (Springer, New York, 1997).
K. S. Brown, Cohomology of Groups (Springer, New York, 1982).
J. L. Bryant, “Approximating Embeddings of Polyhedra in Codimension Three,” Trans. Am. Math. Soc. 170, 85–95 (1972).
J. L. Bryant, “Triangulation and General Position of PL Diagrams,” Topol. Appl. 34, 211–233 (1990).
S. Buoncristiano, C. P. Rourke, and B. J. Sanderson, A Geometric Approach to Homology Theory (Cambridge Univ. Press, Cambridge, 1976), LMS Lect. Note Ser. 18.
O. Cornea, G. Lupton, J. Oprea, and D. Tanré, Lusternik-Schnirelmann Category (Am. Math. Soc., Providence, RI, 2003), Math. Surv. Monogr. 103.
M. C. Crabb, ℤ/2-Homotopy Theory (Cambridge Univ. Press, Cambridge, 1980), LMS Lect. Note Ser. 44.
P. E. Conner and E. E. Floyd, “Fixed Point Free Involutions and Equivariant Maps,” Bull. Am. Math. Soc. 66, 416–441 (1960).
M. de Longueville, “Bier Spheres and Barycentric Subdivision,” J. Comb. Theory, Ser. A 105, 355–357 (2004).
J.-P. Dax, “Étude homotopique des espaces de plongements,” Ann. Sci. Éc. Norm. Supér., Sér. 4, 5, 303–377 (1972).
P. J. Eccles and M. Grant, “Bordism Groups of Immersions and Classes Represented by Self-intersections,” Algebr. Geom. Topol. 7, 1081–1097 (2007); arXiv:math/0504152.
A. Flores, “Über n-dimensionale Komplexe, die im R 2n+1 absolut selbstverschlungen sind,” Ergeb. Math. Kolloq. 6, 4–7 (1935).
M. H. Freedman, V. S. Krushkal, and P. Teichner, “Van Kampen’s Embedding Obstruction Is Incomplete for 2-Complexes in ℝ4,” Math. Res. Lett. 1, 167–176 (1994).
D. Gonçalves and A. Skopenkov, “Embeddings of Homology Equivalent Manifolds with Boundary,” Topol. Appl. 153, 2026–2034 (2006).
B. Grünbaum, “Imbeddings of Simplicial Complexes,” Comment. Math. Helv. 44, 502–513 (1969).
A. Haefliger and M. W. Hirsch, “Immersions in the Stable Range,” Ann. Math., Ser. 2, 75, 231–241 (1962).
L. S. Harris, “Intersections and Embeddings of Polyhedra,” Topology 8, 1–26 (1969).
A. Hatcher and F. Quinn, “Bordism Invariants of Intersections of Submanifolds,” Trans. Am. Math. Soc. 200, 327–344 (1974).
H. Hauschild, “Äquivariante Homotopie. I,” Arch. Math. 29, 158–165 (1977).
Sze-Tsen Hu, “Isotopy Invariants of Topological Spaces,” Proc. R. Soc. London, Ser. A 255, 331–366 (1960).
I. M. James, “On Category, in the Sense of Lusternik-Schnirelmann,” Topology 17, 331–348 (1978).
J. R. Klein, “On Embeddings in the Sphere,” Proc. Am. Math. Soc. 133, 2783–2793 (2005); arXiv:math/0310236.
J. R. Klein and B. Williams, “Homotopical Intersection Theory. II: Equivariance,” Math. Z., doi: 10.1007/s00209-009-0491-1 (2009); arXiv: 0803.0017.
V. S. Krushkal, “Embedding Obstructions and 4-Dimensional Thickenings of 2-Complexes,” Proc. Am. Math. Soc. 128, 3683–3691 (2000); arXiv:math/0004058.
Cz. Kosniowski, “Equivariant Cohomology and Stable Cohomotopy,” Math. Ann. 210, 83–104 (1974).
L. Lovász and A. Schrijver, “A Borsuk Theorem for Antipodal Links and a Spectral Characterization of Linklessly Embeddable Graphs,” Proc. Am. Math. Soc. 126, 1275–1285 (1998).
V. O. Manturov, “A Proof of Vassiliev’s Conjecture on the Planarity of Singular Links,” Izv. Ross. Akad. Nauk, Ser. Mat. 69(5), 169–178 (2005) [Izv. Math. 69, 1025–1033 (2005)].
W. S. Massey and D. Rolfsen, “Homotopy Classification of Higher Dimensional Links,” Indiana Univ. Math. J. 34, 375–391 (1985).
J. Matoušek, Using the Borsuk-Ulam Theorem (Springer, Berlin, 2003).
J. Matoušek, M. Tancer, and U. Wagner, “Hardness of Embedding Simplicial Complexes in ℝd,” arXiv: 0807.0336.
J. P. May, Equivariant Homotopy and Cohomology Theory (Am. Math. Soc., Providence, RI, 1996), CBMS Reg. Conf. Ser. Math. 91.
C. McCrory, “Cobordism Operations and Singularities of Maps,” Bull. Am. Math. Soc. 82, 281–283 (1976).
C. McCrory, “A Characterization of Homology Manifolds,” J. London Math. Soc. 16, 149–159 (1977).
C. McCrory, “Zeeman’s Filtration of Homology,” Trans. Am. Math. Soc. 250, 147–166 (1979).
S. A. Melikhov, “Isotopic and Continuous Realizability of Maps in the Metastable Range,” Mat. Sb. 195(7), 71–104 (2004) [Sb. Math. 195, 983–1016 (2004)].
S. A. Melikhov, “Sphere Eversions and Realization of Mappings,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 247, 159–181 (2004) [Proc. Steklov Inst. Math. 247, 143–163 (2004)]; arXiv:math/0305158.
S. A. Melikhov, Review of V.A. Vassiliev’s paper “First-Order Invariants and First-Order Cohomology for Spaces of Embeddings of Self-intersecting Curves in ℝn,” Math. Rev. MR2179414 (2007i:57014). 3 p.
S. A. Melikhov and E. V. Shchepin, “The Telescope Approach to Embeddability of Compacta,” arXiv:math/0612085.
D. M. Meyer, “ℤ/p-Equivariant Maps between Lens Spaces and Spheres,” Math. Ann. 312, 197–214 (1998).
R. Nikkuni, “The Second Skew-Symmetric Cohomology Group and Spatial Embeddings of Graphs,” J. Knot Theory Ramif. 9, 387–411 (2000).
D. Repovš, W. Rosicki, A. Zastrow, and M. Željko, “Constructing Near-Embeddings of Codimension One Manifolds with Countable Dense Singular Sets,” arXiv: 0803.4251.
D. Repovš and A. B. Skopenkov, “A Deleted Product Criterion for Approximability of Maps by Embeddings,” Topol. Appl. 87, 1–19 (1998).
N. Robertson, P. Seymour, and R. Thomas, “Sachs’ Linkless Embedding Conjecture,” J. Comb. Theory, Ser. B 64, 185–227 (1995).
R. H. Rosen, “Decomposing 3-Space into Circles and Points,” Proc. Am. Math. Soc. 11, 918–928 (1960).
C. Rourke and B. Sanderson, “Homology Stratifications and Intersection Homology,” in Proc. Kirbyfest, Berkeley, CA (USA), June 22–26, 1998 (Geom. Topol. Publ., Coventry, 1999), Geom. Topol. Monogr. 2, pp. 455–472.
K. S. Sarkaria, “Embedding and Unknotting of Some Polyhedra,” Proc. Am. Math. Soc. 100, 201–203 (1987).
K. S. Sarkaria, “A One-Dimensional Whitney Trick and Kuratowski’s Graph Planarity Criterion,” Isr. J. Math. 73, 79–89 (1991).
A. Shapiro, “Obstructions to the Embedding of a Complex in a Euclidean Space. I: The First Obstruction,” Ann. Math., Ser. 2, 66, 256–269 (1957).
R. Shinjo and K. Taniyama, “Homology Classification of Spatial Graphs by Linking Numbers and Simon Invariants,” Topol. Appl. 134, 53–67 (2003).
A. Skopenkov, “Embedding and Knotting of Manifolds in Euclidean Spaces,” in Surveys in Contemporary Mathematics, Ed. by N. Young and Y. Choi (Cambridge Univ. Press, Cambridge, 2008), LMS Lect. Note Ser. 347, pp. 248–342; arXiv:math/0604045.
A. Skopenkov, “A New Invariant and Parametric Connected Sum of Embeddings,” Fundam. Math. 197, 253–269 (2007); arXiv:math/0509621.
M. Skopenkov, “Embedding Products of Graphs into Euclidean Spaces,” Fundam. Math. 179, 191–198 (2003); arXiv: 0808.1199.
M. Skopenkov, “On Approximability by Embeddings of Cycles in the Plane,” Topol. Appl. 134, 1–22 (2003); arXiv: 0808.1187.
S. Stolz, “The Level of Real Projective Spaces,” Comment. Math. Helv. 64, 661–674 (1989).
A. S. Švarc, “The Genus of a Fiber Space. I, II,” Tr. Mosk. Mat. Obshch. 10, 217–272 (1961); 11, 99–126 (1962) [Am. Math. Soc. Transl., Ser. 2, 55, 49—140 (1966)].
K. Taniyama, “Homology Classification of Spatial Embeddings of a Graph,” Topol. Appl. 65, 205–228 (1995).
K. Taniyama, “Higher Dimensional Links in a Simplicial Complex Embedded in a Sphere,” Pac. J. Math. 194, 465–467 (2000).
B. R. Ummel, “Some Examples Relating the Deleted Product to Imbeddability,” Proc. Am. Math. Soc. 31, 307–311 (1972).
E. R. van Kampen, “Komplexe in euklidischen Räumen,” Abh. Math. Semin. Univ. Hamburg 9, 72–78, 152–153 (1932).
V. A. Vassiliev, Topology of Complements of Discriminants (Phasis, Moscow, 1997); Partial Engl. transl.: Complements of Discriminants of Smooth Maps: Topology and Applications, Rev. ed. (Am. Math. Soc., Providence, RI, 1994).
V. A. Vasiliev, “First-Order Invariants and Cohomology of Spaces of Embeddings of Self-intersecting Curves in ℝn,” Izv. Ross. Akad. Nauk, Ser. Mat. 69(5), 3–52 (2005) [Izv. Math. 69, 865–912 (2005)].
A. Yu. Volovikov and E. V. Shchepin, “Antipodes and Embeddings,” Mat. Sb. 196(1), 3–32 (2005) [Sb. Math. 196, 1–28 (2005)].
C. Weber, “Plongements de polyèdres dans le domaine métastable,” Comment. Math. Helv. 42, 1–27 (1967).
Tsen-Teh Wu, “On the mod 2 Imbedding Classes of a Triangulable Compact Manifold,” Sci. Record, New Ser. 2(3), 435–438 (1958).
Wen-Tsün Wu, “On the Realization of Complexes in Euclidean Spaces. I, II, III,” Acta Math. Sinica 5, 505–552 (1955); 7, 79–101 (1957); 8, 79–94 (1958). Engl. transl.: Sci. Sinica 7, 251–297, 365–387 (1958); 8, 133–150 (1959); “Parts I, III,” in Selected Works of Wen-Tsun Wu (World Sci., Hackensack, NJ, 2008), pp. 23–69, 71–83.
I. Yaschenko, “Embedding a Smooth Compact Manifold into ℝn,” in Problems from Topology Atlas, Topology atlas document # qaaa-04 (1996), http://at.yorku.ca/q/a/a/a/04.htm
E. C. Zeeman, “Polyhedral n-Manifolds. II: Embeddings,” in Topology of 3-Manifolds and Related Topics, Ed. by M. K. Fort, Jr. (Prentice-Hall, Englewood Cliffs, NJ, 1962), pp. 64–70.
A. Cavicchioli and L. Grasselli, “Cohomological Products and Transversality,” Rend. Sem. Mat. Torino 40(3), 115–125 (1982).
R. A. Fenn, Techniques of Geometric Topology (Cambridge Univ. Press, Cambridge, 1983), LMS Lect. Note Ser. 57.
R. Fenn and D. Sjerve, “Geometric Cohomology Theory,” Contemp. Math. 20, 79–102 (1983).
L. Grasselli, “Subdivision and Poincaré Duality,” Riv. Mat. Univ. Parma, Ser. 4, 9, 95–103 (1983).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Melikhov, S.A. The van Kampen obstruction and its relatives. Proc. Steklov Inst. Math. 266, 142–176 (2009). https://doi.org/10.1134/S0081543809030092
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543809030092