New results on embeddings of polyhedra and manifolds in Euclidean spaces

D. Repovš   ; A. B. Skopenkov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 54 (1999) no. 6, pp. 1149-1196

Résumé

The aim of this survey is to present several classical results on embeddings and isotopies of polyhedra and manifolds in $\mathbb R^m$. We also describe the revival of interest in this beautiful branch of topology and give an account of new results, including an improvement of the Haefliger–Weber theorem on the completeness of the deleted product obstruction to embeddability and isotopy of highly connected manifolds in $\mathbb R^m$ (Skopenkov) as well as the unimprovability of this theorem for polyhedra (Freedman, Krushkal, Teichner, Segal, Skopenkov, and Spiez) and for manifolds without the necessary connectedness assumption (Skopenkov). We show how algebraic obstructions (in terms of cohomology, characteristic classes, and equivariant maps) arise from geometric problems of embeddability in Euclidean spaces. Several classical and modern results on completeness or incompleteness of these obstructions are stated and proved. By these proofs we illustrate classical and modern tools of geometric topology (engulfing, the Whitney trick, van Kampen and Casson finger moves, and their generalizations).

Bibliographie

[1] Adachi M., Embeddings and Immersions, Transl. of Math. Monographs, 124, Amer. Math. Soc., Providence, RI, 1993 | Zbl

[2] Akhmetiev P., Repovš D., Skopenkov A., “Embedding products of low-dimensional manifolds in $\mathbb R^m$”, Topology Appl. (to appear)

[3] Akhmetiev P., Repovš D., Skopenkov A., “Obstructions to approximating maps of surfaces in $\mathbb R^4$ by embeddings”, Topology Appl., submitted

[4] Akin E., “Manifold phenomena in the theory of polyhedra”, Trans. Amer. Math. Soc., 143 (1969), 413–473 | DOI | MR | Zbl

[5] Arnold V. I., “Problema 6”, Matem. prosveschenie, 1958, no. 3, 273

[6] Arnold V. I., “O predstavlenii nepreryvnykh funktsii trekh peremennykh superpozitsiyami nepreryvnykh funktsii dvukh peremennykh”, Matem. sb., 48:1 (1959), 3–74 ; 56:3 (1962), 392 | MR | Zbl

[7] Boechat J., “Plongements différentiables de variétés de dimension $4k$ dans $\mathbb R^{6k+1}$”, Comment Math. Helv., 46 (1971), 141–161 | DOI | MR | Zbl

[8] Boechat J., Haefliger A., “Plongements differentiables de varietes orientees de dimension 4 dans $\mathbb R^7$”, Essays on Topology and Related Topics, Mem. dedié à G. de Rham, Springer-Verlag, New York, 1970, 156–166 | MR

[9] Borsuk K., “Über stetige Abbildungen der euklidischen Räume”, Fund. Math., 21 (1933), 236–246 | MR

[10] Brodsky N., Repovš D., Skopenkov A. B., “A classification of 3-thickenings of 2-polyhedra”, Topology Appl., 94 (1999), 307–314 | DOI | MR

[11] Brown R. L., “Immersions and embeddings up to cobordism”, Canad. J. Math., 23 (1971), 1102–1115 | MR

[12] Bryant J. L., “Approximating embeddings of polyhedra in codimension 3”, Trans. Amer. Math. Soc., 170 (1972), 85–95 | DOI | MR | Zbl

[13] Cappell S. E., Shaneson J. L., “Imbeddings and immersions of 4-dimensional manifolds in $\mathbb R^6$”, Geometric Topology, ed. J. C. Cantrell, Academic Press, New York, 1979, 301–303 | MR

[14] Casson A. J., “Three lectures on new infinite constructions in 4-dimensional manifolds”, A la Récherche de la Topologie Perdue, Progr. Math., 62, eds. L. Guillou, A. Marin, Birkhäuser, Boston, 1986, 201–244; V poiskakh utrachennoi topologii, Sb., eds. L. Giiu, A. Maren, Mir, M., 1989 | MR

[15] Cavicchioli A., Repovš D., Skopenkov A. B., “Open problems on graphs, arising from geometric topology”, Topology Appl., 84 (1998), 207–226 | DOI | MR | Zbl

[16] Chernavskii A. V., “Topologicheskie vlozheniya poliedrov v evklidovy prostranstva”, Dokl. AN SSSR, 6 (1965), 1606–1610

[17] Chernavskii A. V., “Kusochno-lineinye approksimatsii vlozhenii kletok i sfer v korazmernosti bolshe dvukh”, Matem. sb., 80 (1969), 339–364 | MR

[18] Claytor S., “Topological immersions of peanian continua in a spherical surface”, Ann. of Math. (2), 35 (1934), 809–835 | DOI | MR | Zbl

[19] Claytor S., “Peanian continua not embeddable in a spherical surface”, Ann. of Math. (2), 38 (1937), 631–646 | DOI | MR | Zbl

[20] Cohen M. M., “A general theory of relative regular neighbourhoods”, Trans. Amer. Math. Soc., 136 (1969), 189–230 | DOI | MR

[21] Cohen R. L., “The immersion conjecture for differentiable manifolds”, Ann. of Math. (2), 122 (1985), 237–328 | DOI | MR | Zbl

[22] Conner P. E., Floyd E. E., “Fixed points free involutions and equivariant maps”, Bull. Amer. Math. Soc., 66 (1960), 416–441 | DOI | MR | Zbl

[23] Daverman R. J., Decompositions of Manifolds, Academic Press, Orlando, Fla., 1986 | Zbl

[24] Donaldson S. K., “The orientation of Yang–Mills moduli spaces and 4-manifold topology”, J. Differential Geom., 26 (1987), 397–428 | MR | Zbl

[25] Dranišnikov A. N., Repovš D., Ščepin E. V., “On intersection of compacta in Euclidean space: the metastable case”, Tsukuba J. Math., 17 (1993), 549–564 | MR | Zbl

[26] Edwards R. D., “The equivalence of close piecewise linear embeddings”, Gen. Topology Appl., 5 (1975), 147–180 | DOI | MR | Zbl

[27] Fang F., “Embedding four manifolds in $\mathbb R^7$”, Topology, 33 (1994), 447–454 | DOI | MR | Zbl

[28] Flores A., “Über $n$-dimensionale Komplexe die im $E^{2n+1}$ absolute Selbstverschlungen sind”, Ergeb. Math. Koll., 6 (1934), 4–7

[29] Freedman M. H., Krushkal V. S., Teichner P., “Van Kampen's embedding obstruction is incomplete for 2-complexes in $\mathbb R^4$”, Math. Res. Lett., 1 (1994), 167–176 | MR | Zbl

[30] Freedman M. H., Quinn F., Topology of 4-Manifolds, Princeton Univ. Press, Princeton, 1990 | Zbl

[31] Fuks D. B., Shvarts A. S., “Tsiklicheskie stepeni poliedrov i problema vlozheniya”, Dokl. AN SSSR, 125 (1959), 285–288 | MR | Zbl

[32] Galecki M., On embeddability of CW-complexes in Euclidean space, Preprint, Univ. of Tennessee, Knoxville, 1992

[33] Gillman D., Matveev S. V., Rolfsen D., “Collapsing and reconstruction of manifolds”, Contemp. Math., 164 (1994), 35–39 | MR | Zbl

[34] Gillman D., Rolfsen D., “Three-manifolds embed in small 3-complexes”, Internat. J. Math., 3 (1992), 179–183 | DOI | MR | Zbl

[35] Gitler S., “Immersion and embedding of manifolds”, Proc. Sympos. Pure Math., 22, 1971, 87–96 | MR | Zbl

[36] Gluck H., “Unknotting $S^1$ in $S^4$”, Bull. Amer. Math. Soc., 69 (1963), 91–94 | DOI | MR | Zbl

[37] Gluck H., “Geometric characterisation of differentiable manifolds in Euclidean space, II”, Michigan Math. J., 15 (1968), 33–50 | DOI | MR | Zbl

[38] Gordon C. M. A., “Embeddings piecewise linear manifolds with boundary”, Proc. Cambridge Philos. Soc., 72 (1972), 21–25 | DOI | MR | Zbl

[39] Gromov M., Differentsialnye sootnosheniya s chastnymi proizvodnymi, Mir, M., 1990

[40] Habegger N., Kaiser U., “Link homotopy in the 2-metastable range”, Topology, 37 (1998), 75–94 | DOI | MR | Zbl

[41] Hacon D. D. J., “Embeddings of $S^p$ in $S^1\times S^q$ in the metastable range”, Topology, 7 (1968), 1–10 | DOI | MR | Zbl

[42] Haefliger A., “Plongements différentiables de variétés dans variétés”, Comment. Math. Helv., 36 (1961), 47–82 | DOI | MR | Zbl

[43] Haefliger A., “Knotted $(4k-1)$-spheres in $6k$-space”, Ann. of Math. (2), 75 (1962), 452–466 | DOI | MR | Zbl

[44] Haefliger A., “Plongements de variétés dans le domaine stable”, Sem. Bourbaki, 1962/63, 15, no. 245, 1964 | Zbl

[45] Haefliger A., “Differentiable links”, Topology, 1 (1962), 241–244 | DOI | MR | Zbl

[46] Haefliger A., “Plongements différentiables dans le domaine stable”, Comment. Math. Helv., 37 (1962), 155–176 | DOI | MR | Zbl

[47] Haefliger A., “Differentiable embeddings of $S^n$ in $S^{n+q}$ for $q>2$”, Ann. of Math. (2), 83 (1966), 402–436 | DOI | MR | Zbl

[48] Haefliger A., “Differentiable embeddings of $S^n$”, Bull. Amer. Math. Soc., 67 (1961), 109–112 | DOI | MR | Zbl

[49] Haefliger A., “Lissage des immersions, I”, Topology, 6 (1967), 221–239 | DOI | MR | Zbl

[50] Haefliger A., Lissage des immersions, II, Preprint, 1966

[51] Haefliger A., “Knotted spheres and related geometric problems”, Proc. Internat. Congr. Math. (Moscow, 1966), 1968, 437–445 | Zbl

[52] Haefliger A., Hirsch M. W., “Immersions in the stable range”, Ann. of Math. (2), 75 (1962), 231–241 | DOI | MR | Zbl

[53] Haefliger A., Hirsch M. W., “On the existence and classification of differentiable embeddings”, Topology, 2 (1963), 129–135 | DOI | MR | Zbl

[54] Haefliger A., Poénaru V., “La classification des immersions combinatoires”, Inst. Hautes Etudes Sci. Publ. Math., 23 (1964), 75–91 | DOI | MR

[55] Harris L. S., “Intersections and embeddings of polyhedra”, Topology, 8 (1969), 1–26 | DOI | MR | Zbl

[56] Hirsch M. W., “Immersions of manifolds”, Trans. Amer. Math. Soc., 93 (1959), 242–276 | DOI | MR | Zbl

[57] Hirsch M. W., “The embedding of bounding manifolds in Euclidean space”, Ann. of Math., 74 (1961), 494–497 | DOI | MR | Zbl

[58] Hirsch M. W., “On embedding 4-manifolds in $\mathbb R^7$”, Proc. Cambridge Philos. Soc., 61 (1965), 657–658 | DOI | MR | Zbl

[59] Hirsch M. W., Zeeman E. C., “Engulfing”, Bull. Amer. Math. Soc., 72 (1966), 113–115 | DOI | MR | Zbl

[60] Horvatič K., “On embedding polyhedra and manifolds”, Trans. Amer. Math. Soc., 157 (1971), 417–436 | DOI | MR | Zbl

[61] Hsiang W. C., Levine J., Szczarba R. H., “On the normal bundle of a homotopy sphere embedded in Euclidean space”, Topology, 3 (1965), 173–181 | DOI | MR | Zbl

[62] Khu S. Ts., Teoriya gomotopii, Mir, M., 1964

[63] Hu S. T., “Isotopy invariants of topological spaces”, Proc. Roy. Soc. London Ser. A, 255 (1960), 331–366 | DOI | MR | Zbl

[64] Hudson J. F. P., “Knotted tori”, Topology, 2 (1963), 11–22 | DOI | MR | Zbl

[65] Hudson J. F. P., “Non-embedding theorem”, Topology, 2 (1963), 123–128 | DOI | MR | Zbl

[66] Hudson J. F. P., “Extending piecewise linear isotopies”, Proc. London Math. Soc. (3), 16 (1966), 651–668 | DOI | MR | Zbl

[67] Hudson J. F. P., “PL embeddings”, Ann. of Math. (2), 85:1 (1967), 1–31 | DOI | MR | Zbl

[68] Hudson J. F. P., “On sunny collapsing”, Proc. Cambridge Philos. Soc., 66 (1969), 247–250 | DOI | MR | Zbl

[69] Hudson J. F. P., Piecewise Linear Topology, Benjamin, New York, 1969 | Zbl

[70] Hudson J. F. P., “Concordance, isotopy and diffeotopy”, Ann. of Math. (2), 91 (1970), 425–448 | DOI | MR | Zbl

[71] Hudson J. F. P., “Obstructions to embedding disks”, Topology of Manifolds, Proc. Univ. Georgia, 1969, 1971, 407–415 | Zbl

[72] Hudson J. F. P., “Embeddings of bounded manifolds”, Proc. Cambridge Philos. Soc., 72 (1972), 11–20 | DOI | MR | Zbl

[73] Hudson J. F. P., Lickorish W. B. R., “Extending piecewise linear concordances”, Quart. J. Math. (2), 22 (1971), 1–12 | DOI | MR | Zbl

[74] Hudson J. F. P., Zeeman E. C., “On combinatorial isotopy”, Inst. Hautes Etudes Sci. Publ. Math., 19 (1964), 69–94 | DOI | Zbl

[75] Hudson J. F. P., Zeeman E. C., “On regular neighborhoods”, Proc. London Math. Soc. (3), 14 (1964), 719–745 ; Correction, 21 (1970), 513–524 | DOI | MR | Zbl | DOI | MR | Zbl

[76] Gurevich V., Volmen G., Teoriya razmernosti, IL, M., 1948

[77] Husch L. S., “$\varepsilon$-maps and embeddings”, Res. Exp. Math., 16 (1988), 273–280 | MR | Zbl

[78] Irwin M. C., “Embeddings of polyhedral manifolds”, Ann. of Math. (2), 82 (1965), 1–14 | DOI | MR | Zbl

[79] James I. M., The Topology of Stiefel Manifolds, London Math. Soc., London, 1976

[80] van Kampen E. R., “Komplexe in euklidischen Räumen”, Abh. Math. Sem. Univ. Hamburg, 9 (1932), 72–78; 152–153 | Zbl

[81] Kearton C., “Obstructions to embeddings and isotopy in the metastable range”, Math. Ann., 243 (1979), 103–113 | DOI | MR | Zbl

[82] Kervaire M. A., “An interpretation of G. Whitehead's generalization of H. Hopf's invariant”, Ann. of Math. (2), 69 (1959), 345–365 | DOI | MR | Zbl

[83] Kervaire M. A., “Smooth homology spheres and their fundamental groups”, Trans. Amer. Math. Soc., 144 (1969), 67–72 | DOI | MR | Zbl

[84] Kervaire M. A., Milnor J. W., “On 2-spheres in 4-manifolds”, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 1651–1657 | DOI | MR | Zbl

[85] Kirby R., “4-manifold problems”, Contemp. Math., 35 (1984), 513–528 | MR | Zbl

[86] Kirby R. C., The Topology of 4-Manifolds, Lecture Notes in Math., 1374, Springer-Verlag, Berlin, 1989 | Zbl

[87] Kolmogorov A. N., “O predstavlenii nepreryvnykh funktsii neskolkikh peremennykh v vide superpozitsii nepreryvnykh funktsii odnogo peremennogo i slozheniya”, Dokl. AN SSSR, 114:5 (1957), 953–956 | MR

[88] Koschorke U., “Link homotopy with many components”, Topology, 30:2 (1991), 267–281 | DOI | MR | Zbl

[89] Kuratowski K., “Sur le problème des courbes gauches en topologie”, Fund. Math., 15 (1930), 271–283

[90] Kurlin V., “Basic embeddings into products of graphs”, Topology Appl., 1999 (to appear)

[91] Kurlin V., Skopenkov A., “Bazisnye vlozheniya grafov v ploskost”, 9-ya letnyaya konferentsiya Turnira Gorodov, 1998, 34–44; Р�Р·Рґ-РІРѕ МЦНМО, 106–113

[92] Lackenby M., “The Whitney trick”, Topology Appl., 71 (1996), 115–118 | DOI | MR | Zbl

[93] Lannes J., “La conjecture des immersions”, Astérisque, 92/93, 1982, 331–346 | MR | Zbl

[94] Levin M., “Dimension and superposition of continuous functions”, Israel J. Math., 70:2 (1990), 205–218 | DOI | MR | Zbl

[95] Levine J., “Unknotting spheres in codimension 2”, Topology, 4 (1965), 9–16 | DOI | MR | Zbl

[96] Lickorish W. B. R., “The piecewise linear unknotting of cones”, Topology, 4 (1965), 67–91 | DOI | MR | Zbl

[97] Lickorish W. B. R., Siebenmann L. C., “Regular neighborhoods and the stable range”, Trans. Amer. Math. Soc., 139 (1969), 207–230 | DOI | MR | Zbl

[98] Mardešić S., Segal J., “A note on polyhedra embeddable in the plane”, Duke Math. J., 33 (1966), 633–638 | DOI | MR | Zbl

[99] Mardešić S., Segal J., “$\varepsilon$-mappings and generalized manifolds”, Michigan Math. J., 14 (1967), 171–182 | DOI | MR | Zbl

[100] Massey W. S., “On the Stiefel–Whitney classes of a manifold, 1”, Amer. J. Math., 82 (1960), 92–102 | DOI | MR | Zbl

[101] Massey W. S., “On the Stiefel–Whitney classes of a manifold, 2”, Proc. Amer. Math. Soc., 13 (1962), 938–942 | DOI | MR | Zbl

[102] Massey W., “Homotopy classification of 3-component links of codimension greater than 2”, Topology Appl., 34 (1990), 269–300 | DOI | MR | Zbl

[103] McLane S., Adkisson V. W., “Extensions of homeomorphisms on the spheres”, Lectures on Topology, Univ. of Michigan Press, Ann. Arbor MI, 1941, 223–230

[104] Miller R. T., “Close isotopies on piecewise-linear manifolds”, Trans. Amer. Math. Soc., 151 (1970), 597–628 | DOI | MR | Zbl

[105] Miller R. T., “Approximating codimension 3 embeddings”, Ann. of Math. (2), 95 (1972), 406–416 | DOI | MR | Zbl

[106] Millett K. C., “Piecewise linear embeddings of manifolds”, Illinois J. Math., 19 (1975), 354–369 | MR | Zbl

[107] Milnor Dzh., Stasheff Dzh., Kharakteristicheskie klassy, Mir, M., 1979 | Zbl

[108] Neuwirth L., “An algorithm for the construction of 3-manifolds from 2-complexes”, Proc. Cambridge Philos. Soc., 64 (1968), 603–613 | DOI | MR | Zbl

[109] Novikov S. P., “Vlozheniya odnosvyaznykh mnogoobrazii v evklidovy prostranstva”, Dokl. AN SSSR, 138 (1961), 775–778 | MR | Zbl

[110] Ostrand P. A., “Dimension of metric spaces and Hilbert's problem 13”, Bull. Amer. Math. Soc., 7 (1965), 619–622 | DOI | MR

[111] Paechter G. F., “On the groups $\pi_r(V_{mn})$. I–V”, Quart. J. Math. Oxford Ser. (2), 7 (1956), 249–265 ; 9 (1958), 8–27 ; 10 (1959), 17–37 ; 10 (1959), 241–260 ; 11 (1960), 1–16 | DOI | MR | DOI | MR | Zbl | DOI | MR | DOI | DOI | MR

[112] Penrose R., Whitehead J. H. C., Zeeman E. C., “Embeddings of manifolds in a Euclidean space”, Ann. of Math. (2), 73 (1961), 613–623 | DOI | MR | Zbl

[113] Poénaru V., “Regular homotopy and isotopy”, Mimeographed notes, Harvard Univ., 1964

[114] Pontryagin L. S., “Kharakteristicheskie tsikly gladkikh mnogoobrazii”, Dokl. AN SSSR, 35:2 (1942), 35–39 | MR

[115] Postnikov M. M., Teoriya gomotopii kletochnykh prostranstv, Nauka, M., 1985 | Zbl

[116] Rees E., “Some embeddings of Lie groups in Euclidean spaces”, Mathematika (London), 18 (1971), 152–156 | MR | Zbl

[117] Rees E., “Problems concerning embeddings of manifolds”, Adv. in Math. (China), 19 (1990), 72–79 | MR | Zbl

[118] Repovš D., Skopenkov A. B., “Embeddability and isotopy of polyhedra in Euclidean spaces”, Proc. Steklov Inst. Math., 212, 1996, 173–188

[119] Repovš D., Skopenkov A. B., “A deleted product criterion for approximability of a map by embeddings”, Topology Appl., 87 (1998), 1–19 | DOI | MR | Zbl

[120] Repovš D., Skopenkov A. B., “Borromean rings and embedding obstructions”, Proc. Math. Inst. Russ. Acad. Sci. (to appear) | Zbl

[121] Repovš D., Skopenkov A. B., Projected embeddings and desuspension of $\alpha$-invariant, Preprint | Zbl

[122] Repovš D., Skopenkov A. B., Ščepin E. V., “On embeddability of $X\times I$ into Euclidean space”, Houston J. Math., 21 (1995), 199–204 | MR | Zbl

[123] Rokhlin V. A., “Vlozhenie neorientiruemykh trekhmernykh mnogoobrazii v pyatimernoe evklidovo prostranstvo”, Dokl. AN SSSR, 160 (1965), 549–551 | MR | Zbl

[124] Rourke C. P., Sanderson B. J., “Block bundles, I”, Ann. of Math. (2), 87 (1968), 1–28 | DOI | MR | Zbl

[125] Rurk K. P., Sanderson B. Dzh., Vvedenie v kusochno-lineinuyu topologiyu, Mir, M., 1974

[126] Rushing T. B., Topological Embeddings, Academic Press, New York, 1973 | Zbl

[127] Sarkaria K. S., “A one-dimensional Whitney trick and Kuratowski's graph planarity criterion”, Israel J. Math., 73 (1991), 79–89 | DOI | MR | Zbl

[128] Sarkaria K. S., “Kuratowski complexes”, Topology, 30 (1991), 67–76 | DOI | MR | Zbl

[129] Schepin E. V., Shtanko M. A., “Spektralnyi kriterii vlozhimosti kompaktov v evklidovy prostranstva”, Trudy Leningr. mezhdun. topol. konf., Nauka, L., 1983, 135–142

[130] Segal J., Skopenkov A., Spie. z S., “Embeddings of polyhedra in $\mathbb R^m$ and the deleted product obstruction”, Topology Appl., 85 (1998), 335–344 | DOI | MR | Zbl

[131] Segal J., Spie. z S., “Quasi embeddings and embeddings of polyhedra in $\mathbb R^m$”, Topology Appl., 45 (1992), 275–282 | DOI | MR | Zbl

[132] Shapiro A., “Obstructions to the embedding of a complex in a Euclidean space. I: The first obstruction”, Ann. of Math. (2), 66 (1957), 256–269 | DOI | MR | Zbl

[133] Skopenkov A., “Geometricheskoe dokazatelstvo teoremy Noivirta ob utolschaemosti 2-mernykh poliedrov”, Matem. zametki, 56:2 (1994), 94–98 | MR

[134] Skopenkov A. B., “A description of continua basically embeddable in $\mathbb R^2$”, Topology Appl., 65 (1995), 29–48 | DOI | MR | Zbl

[135] Skopenkov A. B., “On the deleted product criterion for embeddability of manifolds in $\mathbb R^m$”, Comment. Math. Helv., 72 (1997), 543–555 | DOI | MR | Zbl

[136] Skopenkov A. B., “On the deleted product criterion for embeddability in $\mathbb R^m$”, Proc. Amer. Math. Soc., 126 (1998), 2467–2476 | DOI | MR | Zbl

[137] Skopenkov A., “On the deleted product criterion for embeddings and immersions of manifolds in $\mathbb R^m$”, Topology, submitted

[138] Skopenkov A., “On the generalized Massey–Rolfsen invariant for link maps in $\mathbb R^m$”, Fund. Math., submitted

[139] Smale S., “The classification of immersions of spheres in Euclidean spaces”, Ann. of Math. (2), 69 (1959), 327–344 | DOI | MR | Zbl

[140] Spie. z S., Toruńczyk H., “Moving compacta in $\mathbb R^m$ apart”, Topology Appl., 41 (1991), 193–204 | DOI | MR | Zbl

[141] Stallings J., “Polyhedral homotopy-spheres”, Bull. Amer. Math. Soc., 66 (1960), 485–488 | DOI | MR | Zbl

[142] Stallings J., “On topologically unknotted spheres”, Ann. of Math. (2), 77 (1963), 490–503 | DOI | MR | Zbl

[143] Stallings J., “Homology and central series of groups”, J. Algebra, 2 (1965), 170–181 | DOI | MR | Zbl

[144] Sternfeld Y., “Dimension, superposition of functions and separation of points in compact metric spaces”, Israel J. Math., 50 (1985), 13–53 | DOI | MR | Zbl

[145] Sternfeld Y., “Hilbert's 13th problem and dimension”, Lecture Notes in Math., 1376, 1989, 1–49 | MR | Zbl

[146] Szücs A., “The Gromov–Eliashberg proof of Haefliger's theorem”, Studia Sci. Math. Hungar., 17 (1982), 303–318 | MR | Zbl

[147] Taniyama K., “Homology classification of spatial embeddings of a graph”, Topology Appl., 65 (1995), 205–228 | DOI | MR | Zbl

[148] Toda Kh., Kompozitsionnye metody v teorii gomotopicheskikh grupp sfer, Nauka, M., 1982 | Zbl

[149] Vasilev V. A., Topologiya dopolnenii k diskriminantam, Fazis, M., 1997

[150] Wall C. T. C., “All 3-manifolds imbed in 5-space”, Bull. Amer. Math. Soc., 71 (1965), 490–503 | DOI

[151] Wall C. T. C., “Classification problems in differential topology. IV: Thickenings”, Topology, 5 (1966), 73–94 | DOI | MR | Zbl

[152] Wall C. T. C., Surgery on Compact Manifolds, Academic Press, London, 1970

[153] Weber C., “Plongements de polyèdres dans le domaine metastable”, Comment. Math. Helv., 42 (1967), 1–27 | DOI | MR | Zbl

[154] Whitehead J. H. C., “Simplicial spaces, nuclei and $m$-groups”, Proc. London Math. Soc., 45 (1939), 243–327 | DOI | MR | Zbl

[155] Whitehead J. H. C., “On the groups $\pi_r(V_m,V_n)$ and sphere bundles”, Proc. London Math. Soc. (2), 48 (1944), 243–291 ; correction, 49 (1945), 479–481 | DOI | MR | DOI | MR | Zbl

[156] Whitney H., “Differentiable manifolds in Euclidean space”, Proc. Nat. Acad. Sci. U.S.A., 21:7 (1935), 462–464 | DOI | Zbl

[157] Whitney H., “On the topology of differentiable manifolds”, Lectures on Topology, Univ. of Michigan Press, Ann Arbor, MI, 1941, 101–141 | MR | Zbl

[158] Whitney H., “The self-intersections of a smooth $n$-manifolds in $2n$-space”, Ann. of Math. (2), 45 (1944), 220–246 | DOI | MR | Zbl

[159] Wu W. T., “On the realization of complexes in Euclidean spaces. I; II; III”, Sci. Sinica, 7 (1958), 251–297 ; 365–387; 8 (1959), 133–150 | MR | Zbl | MR

[160] Wu W. T., A Theory of Embedding, Immersion and Isotopy of Polytopes in an Euclidean Space, Science Press, Peking, 1965

[161] Yang C. T., “On theorems of Borsuk–Ulam, Kakutani–Yamabe–Yujobo and Dyson, I”, Ann. of Math. (2), 60 (1954), 262–282 | DOI | MR | Zbl

[162] Zeeman E. C., “Unknotting spheres”, Ann. of Math. (2), 72 (1960), 350–360 | DOI | MR

[163] Zeeman E. C., “Isotopies and knots in manifolds”, Topology of 3-Manifolds and Related Topics, ed. M. K. Fort, Prentice-Hall, Englewood Cliffs, 1962

[164] Zeeman E. C., “The Poincare conjecture for $n\ge5$”, Topology of 3-Manifolds and Related Topics, ed. M. K. Fort, Prentice-Hall, Englewood Cliffs, 1962

[165] Zeeman E. C., “Unknotting combinatorial balls”, Ann. of Math. (2), 78 (1963), 501–526 | DOI | MR | Zbl

[166] Zeeman E. C., Seminar on Combinatorial Topology, Ch. 1–6, 1963; Ch. 7, 1965; Ch. 8, Inst. Hautes Etudes Sci., Paris, 1966

[167] Zhongmou L., “Every 3-manifold with boundary embeds in $\text{Triod}\times\text{Triod}\times I$”, Proc. Amer. Math. Soc., 122 (1994), 575–579 | DOI | MR