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Lisa
Lisa
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Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. Question: Can I always find a smooth homotopy $f:M_1 \times I \rightarrow M_2$$F:M_1 \times I \rightarrow M_2$ such that $p$ is a regular value for $F$?

This question came up in the smooth manifolds course I'm teaching right now. If it had a positive (and relatively elementary) answer, then it would allow me to greatly simplify some of the proofs I'm getting ready to present.

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. Question: Can I always find a smooth homotopy $f:M_1 \times I \rightarrow M_2$ such that $p$ is a regular value for $F$?

This question came up in the smooth manifolds course I'm teaching right now. If it had a positive (and relatively elementary) answer, then it would allow me to greatly simplify some of the proofs I'm getting ready to present.

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. Question: Can I always find a smooth homotopy $F:M_1 \times I \rightarrow M_2$ such that $p$ is a regular value for $F$?

This question came up in the smooth manifolds course I'm teaching right now. If it had a positive (and relatively elementary) answer, then it would allow me to greatly simplify some of the proofs I'm getting ready to present.

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Lisa
Lisa
  • 321
  • 1
  • 7

Homotopies with prescribed regular values

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. Question: Can I always find a smooth homotopy $f:M_1 \times I \rightarrow M_2$ such that $p$ is a regular value for $F$?

This question came up in the smooth manifolds course I'm teaching right now. If it had a positive (and relatively elementary) answer, then it would allow me to greatly simplify some of the proofs I'm getting ready to present.