Lifting generic maps to embeddings. The double point obstruction
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2017
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| _version_ | 1866909940521631744 |
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| author | Melikhov, Sergey A. |
| author_facet | Melikhov, Sergey A. |
| contents | Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $\{(x,y)\in N\times N\mid f(x)=f(y),\,x\ne y\}$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen and obtain some other applications.
We also discuss several criteria for lifting of a non-degenerate PL map or a $C^0$-stable smooth map $f:N^n\to M^m$, where $m\ge n$, to an embedding in $M\times\mathbb R$, elaborating on V. Poénaru's observations. In particular, the existence of such a lift is determined by the equivariant homotopy type of the diagram consisting of the three projections from the triple point locus $\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),\,x\ne y\ne z\ne x\}$ to the double point locus.
The three Appendices, which can be read independently of the rest of the paper, are devoted to stable and generic maps. Appendix B introduces an elementary theory of stable PL maps. Appendix C extends the 2-multi-0-jet transversality theorem over the usual compactification of $M\times M\setminusΔ_M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1711_03518 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Lifting generic maps to embeddings. The double point obstruction Melikhov, Sergey A. Geometric Topology Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $\{(x,y)\in N\times N\mid f(x)=f(y),\,x\ne y\}$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen and obtain some other applications. We also discuss several criteria for lifting of a non-degenerate PL map or a $C^0$-stable smooth map $f:N^n\to M^m$, where $m\ge n$, to an embedding in $M\times\mathbb R$, elaborating on V. Poénaru's observations. In particular, the existence of such a lift is determined by the equivariant homotopy type of the diagram consisting of the three projections from the triple point locus $\{(x,y,z)\in N\times N\times N\mid f(x)=f(y)=f(z),\,x\ne y\ne z\ne x\}$ to the double point locus. The three Appendices, which can be read independently of the rest of the paper, are devoted to stable and generic maps. Appendix B introduces an elementary theory of stable PL maps. Appendix C extends the 2-multi-0-jet transversality theorem over the usual compactification of $M\times M\setminusΔ_M$. |
| title | Lifting generic maps to embeddings. The double point obstruction |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/1711.03518 |