Sphere Eversions and Realization of Mappings

S. A. Melikhov

Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 159-181.

Résumé

P. M. Akhmetiev used a controlled version of the stable Hopf invariant to show that any (continuous) map $N\to M$ between stably parallelizable compact $n$-manifolds, $n\ne 1,2,3,7$, is realizable in $\mathbb R^{2n}$, i.e., the composition of $f$ with an embedding $M\subset \mathbb R^{2n}$ is $C^0$-approximable by embeddings. It has been long believed that any degree-$2$ map $S^3\to S^3$ obtained by capping off at infinity a time-symmetric (e.g., Shapiro's) sphere eversion $S^2\times I\to \mathbb R^3$ is nonrealizable in $\mathbb R^6$. We show that there exists a self-map of the Poincaré homology 3-sphere that is nonrealizable in $\mathbb R^6$, but every self-map of $S^n$ is realizable in $\mathbb R^{2n}$ for each $n>2$. The latter, together with a ten-line proof for $n=2$ due essentially to M. Yamamoto, implies that every inverse limit of $n$-spheres embeds in $\mathbb R^{2n}$ for $n>1$, which settles R. Daverman's 1990 problem. If $M$ is a closed orientable 3-manifold, we show that a map $S^3\to M$ that is nonrealizable in $\mathbb R^6$ exists if and only if $\pi _1(M)$ is finite and has even order. As a byproduct, an element of the stable stem $\Pi _3$ with nontrivial stable Hopf invariant is represented by a particularly simple immersion $S^3\looparrowright \mathbb R^4$, namely, by the composition of the universal $8$-covering over $Q^3=S^3/\{\pm 1,\pm i,\pm j,\pm k\}$ and an explicit embedding $Q^3\hookrightarrow \mathbb R^4$.

Bibliographie

[1] Akhmetev P. M., “Ob izotopicheskoi i diskretnoi realizatsiyakh otobrazhenii $n$-sfery v evklidovom prostranstve”, Mat. sb., 187:7 (1996), 3–34 | MR

[2] Akhmetev P. M., “Vlozheniya kompaktov, stabilnye gomotopicheskie gruppy sfer i teoriya osobennostei”, UMN, 55:3 (2000), 3–62 | MR

[3] Akhmet'ev P. M., “Pontrjagin–Thom construction for approximation of mappings by embeddings”, Topol. and Appl., 140 (2004), 133–149 | DOI | MR

[4] Akhmetev P. M., “Geometricheskii podkhod k stabilnym gomotopicheskim gruppam sfer. Invarianty Khopfa–Adamsa”, Izv. RAN. Ser. mat. (to appear)

[5] Akhmetev P. M., Melikhov S. A., “Ob izotopicheskoi realizuemosti nepreryvnykh otobrazhenii”, Zap. nauch. sem. POMI, 276, 2000, 53–87 ; J. Math. Sci. New York, 113 (2003), 759–776 | MR | DOI

[6] Akhmetiev P. M., Szűcs A., “Geometric proof of the easy part of the Hopf invariant one theorem”, Math. Slovaca, 49:1 (1999), 71–74 | MR

[7] Bredon G., Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980 | MR | Zbl

[8] Brown M., “On the inverse limit of Euclidean $n$-spheres”, Trans. Amer. Math. Soc., 96 (1960), 129–134 | DOI | MR | Zbl

[9] Brown R. L. W., “A note on immersions up to cobordism”, Ill. J. Math., 21 (1977), 240–241 | MR | Zbl

[10] Carter J. S., “On generalizing Boy's surface: constructing a generator of the third stable stem”, Trans. Amer. Math. Soc., 298 (1986), 103–122 | DOI | MR | Zbl

[11] Conner P. E., Floyd E. E., “Fixed point free involutions and equivariant maps”, Bull. Amer. Math. Soc., 66:6 (1960), 416–441 | DOI | MR | Zbl

[12] Daverman R., “Problems about finite-dimensional manifolds”, Open problems in topology, eds. van Mill, G. M. Reed, North-Holland, Amsterdam, 1990, 432–455 ; http://www1.elsevier.com/homepage/sac/opit/26/article.pdf | MR

[13] Duvall P. F., Husch L. S., Embedding coverings into bundles with applications, Mem. AMS, 38, no. 263, Amer. Math. Soc., Providence, RI, 1982 | MR

[14] Eccles P. J., “Multiple points of codimension one immersions”, Topology symposium, Siegen, 1979, Lect. Notes Math., 788, Springer, Berlin, 1980, 23–38 | MR

[15] Francis G. K., A topological picturebook, Springer, New York, 1988 ; Fransis Dzh., Knizhka s kartinkami po topologii, Mir, M., 1991 | MR | MR

[16] Freedman M. H., “Quadruple points of $3$-manifolds in $S^4$”, Comment. Math. Helv., 53 (1978), 385–394 | DOI | MR | Zbl

[17] Freedman M. H., Quinn F., Topology of $4$-manifolds, Princeton Univ. Press, Princeton, 1990 | MR | Zbl

[18] Gillman D., “The spinning and twisting of a complex in a hyperplane”, Ann. Math. Ser. 2, 85 (1967), 32–41 | DOI | MR | Zbl

[19] Gromov M., Differentsialnye sootnosheniya s chastnymi proizvodnymi, Mir, M., 1990 | MR

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

[21] Khirsh M., Differentsialnaya topologiya, Mir, M., 1979 | MR | Zbl

[22] Hughes J. F., “Another proof that every eversion of the sphere has a quadruple point”, Amer. J. Math., 107 (1985), 501–505 | DOI | MR | Zbl

[23] Isbell J. R., “Embeddings of inverse limits”, Ann. Math. Ser. 2, 70 (1959), 73–84 | DOI | MR | Zbl

[24] Keesling J. E., Wilson D. C., “Embedding $T^n$-like continua in Euclidean space”, Topol. and Appl., 21 (1985), 241–249 | DOI | MR | Zbl

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

[26] Kervaire M. A., Milnor J. W., “Groups of homotopy spheres, I”, Ann. Math. Ser. 2, 77 (1963), 504–537 | DOI | MR | Zbl

[27] Koschorke U., “Multiple points of immersions and the Kahn–Priddy theorem”, Math. Ztschr., 169 (1979), 223–236 | DOI | MR

[28] Koschorke U., Sanderson B., “Geometric interpretations of the generalized Hopf invariant”, Math. Scand., 41 (1977), 199–217 | MR

[29] Liulevicius A., “Immersions up to cobordism”, Ill. J. Math., 19 (1975), 149–164 | MR | Zbl

[30] Massey W., “Imbeddings of projective planes and related manifolds in spheres”, Indiana Univ. Math. J., 23 (1973/74), 791–812 | DOI | MR

[31] Max N., Banchoff T., “Every sphere eversion has a quadruple point”, Contributions to analysis and geometry, eds. D. N. Clark, G. Pecelli, R. Sachsteder, John Hopkins Univ. Press, Baltimore, 1981, 191–209 | MR

[32] McCord M. C., “Embedding $\mathscr{P}$-like compacta in manifolds”, Canad. J. Math., 19 (1967), 321–332 | MR | Zbl

[33] Miller J. G., “Self-intersections of some immersed manifolds”, Trans. Amer. Math. Soc., 136 (1969), 329–338 | DOI | MR | Zbl

[34] Nowik T., “Quadruple points of regular homotopies of surfaces in $3$-manifolds”, Topology, 39 (2000), 1069–1088 | DOI | MR | Zbl

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

[36] Repovš D., Skopenkov A. B., “A deleted product criterion for approximability of maps by embeddings”, Topol. and Appl., 87 (1998), 1–19 | DOI | MR | Zbl

[37] Rourke C., Sanderson B., “The compression theorem, I”, Geom. and Topol., 5 (2001), 399–429 | DOI | MR | Zbl

[38] Siekł{u}cki K., “Realization of mappings”, Fund. Math., 65:3 (1969), 325–343 | MR | Zbl

[39] Yamamoto M., Lifting a generic map from a surface into the plane to an embedding into $4$-space, Preprint, Dept. Math. Hokkaido Univ., 2004 | MR

[40] McCrory C., “Cobordism operations and singularities of maps”, Bull. Amer. Math. Soc., 82 (1976), 281–283 | DOI | MR | Zbl

[41] McCrory C., “Geometric homology operations”, Studies in algebraic topology, Adv. Math. Suppl. Stud., 5, Acad. Press, New York, 1979, 119–141 | MR