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 Tr. Mat. Inst. Steklova, 2004, Volume 247, Pages 159–181 (Mi tm15)

Sphere Eversions and Realization of Mappings

S. A. Melikhovab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Florida

Abstract: 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 $Q^3\hookrightarrow \mathbb R^4$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 247, 143–163

Bibliographic databases:

UDC: 515.163.6

Citation: S. A. Melikhov, “Sphere Eversions and Realization of Mappings”, Geometric topology and set theory, Collected papers. Dedicated to the 100th birthday of professor Lyudmila Vsevolodovna Keldysh, Tr. Mat. Inst. Steklova, 247, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 159–181; Proc. Steklov Inst. Math., 247 (2004), 143–163

Citation in format AMSBIB
\Bibitem{Mel04} \by S.~A.~Melikhov \paper Sphere Eversions and Realization of Mappings \inbook Geometric topology and set theory \bookinfo Collected papers. Dedicated to the 100th birthday of professor Lyudmila Vsevolodovna Keldysh \serial Tr. Mat. Inst. Steklova \yr 2004 \vol 247 \pages 159--181 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm15} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2168168} \zmath{https://zbmath.org/?q=an:1107.57013} \transl \jour Proc. Steklov Inst. Math. \yr 2004 \vol 247 \pages 143--163 

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This publication is cited in the following articles:
1. P. M. Akhmet'ev, “A Remark on the Realization of Mappings of the 3-Dimensional Sphere into Itself”, Proc. Steklov Inst. Math., 247 (2004), 4–8
2. A. V. Chernavskii, “On the work of L. V. Keldysh and her seminar”, Russian Math. Surveys, 60:4 (2005), 589–614
3. P. M. Akhmet'ev, “Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants”, J. Math. Sci., 159:6 (2009), 753–760
4. Yamamoto M., “Lifting a generic map of a surface into the plane to an embedding into 4–space”, Illinois Journal of Mathematics, 51:3 (2007), 705–721
5. Proc. Steklov Inst. Math., 266 (2009), 142–176
6. Ekholm T., Takase M., “Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions”, Bull London Math Soc, 43:2 (2011), 251–266
7. S. A. Melikhov, “Transverse fundamental group and projected embeddings”, Proc. Steklov Inst. Math., 290:1 (2015), 155–165
8. Takase M., Tanaka K., “Regular-equivalence of 2-knot diagrams and sphere eversions”, Math. Proc. Camb. Philos. Soc., 161:2 (2016), 237–246
9. Skopenkov A., “Stability of Intersections of Graphs in the Plane and the Van Kampen Obstruction”, Topology Appl., 240 (2018), 259–269
10. Rizell G.D., Golovko R., “The Stable Morse Number as a Lower Bound For the Number of Reeb Chords”, J. Symplectic Geom., 16:5 (2018), 1209–1248
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