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Tr. Mat. Inst. Steklova, 2015, Volume 290, Pages 166–177 (Mi tm3646)  

Transverse fundamental group and projected embeddings

S. A. Melikhov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: For a generic degree $d$ smooth map $f:N^n\to M^n$ we introduce its “transverse fundamental group” $\pi (f)$, which reduces to $\pi _1(M)$ in the case where $f$ is a covering, and in general admits a monodromy homomorphism $\pi (f)\to S_{|d|}$; nevertheless, we show that $\pi (f)$ can be nontrivial even for rather simple degree $1$ maps $S^n\to S^n$. We apply $\pi (f)$ to the problem of lifting $f$ to an embedding $N\hookrightarrow M\times \mathbb R^2$: for such a lift to exist, the monodromy $\pi (f)\to S_{|d|}$ must factor through the group of concordance classes of $|d|$-component string links. At least if $|d|<7$, this requires $\pi (f)$ to be torsion-free.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.1134/S0371968515030140

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 290:1, 155–165

Bibliographic databases:

ArXiv: 1505.00505
UDC: 515.162.6+515.143.3
Received: March 15, 2015

Citation: S. A. Melikhov, “Transverse fundamental group and projected embeddings”, Modern problems of mathematics, mechanics, and mathematical physics, Collected papers, Tr. Mat. Inst. Steklova, 290, MAIK Nauka/Interperiodica, Moscow, 2015, 166–177; Proc. Steklov Inst. Math., 290:1 (2015), 155–165

Citation in format AMSBIB
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\pages 166--177
\publ MAIK Nauka/Interperiodica
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  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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