Difference between revisions of "Compactified Morse trajectory spaces"

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* Introduce smooth evaluation maps <math>{\rm ev}^\pm: \overline\mathcal{M}(\cdot,\cdot) \to L</math>,  
 
* Introduce smooth evaluation maps <math>{\rm ev}^\pm: \overline\mathcal{M}(\cdot,\cdot) \to L</math>,  
 
* Define the length <math>\ell: \overline \mathcal{M}(L,L) \to [0,\infty]</math> by <math>\ell(\gamma)= a</math> for <math>\gamma: [0,a] \to L</math> and <math>\ell(\underline{\gamma})=\infty</math> for all generalized (''broken'') Morse trajectories <math>\underline{\gamma}</math>
 
* Define the length <math>\ell: \overline \mathcal{M}(L,L) \to [0,\infty]</math> by <math>\ell(\gamma)= a</math> for <math>\gamma: [0,a] \to L</math> and <math>\ell(\underline{\gamma})=\infty</math> for all generalized (''broken'') Morse trajectories <math>\underline{\gamma}</math>
 +
* discuss boundary&corner stratification, in particular note that <math>L\subset \partial^1\overline\mathcal{M}(L,L)</math> (the set of trajectories with <math>\ell=0</math>) is isolated from all other boundary strata (made up of generalized trajectories with <math>\ell=\infty</math>)

Revision as of 16:32, 6 June 2017

table of contents

Consider a smooth manifold L equipped with a Morse function f:L\to \mathbb{R} and a metric so that the gradient vector field \nabla f satisfies the Morse-Smale conditions. Then the Morse trajectory spaces

{\begin{alignedat}{4}{\mathcal {M}}(L,L)&=\{\gamma :&[0,a]\;\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),a\geq 0\},\\{\mathcal {M}}(p^{-},L)&=\{\gamma :&(-\infty ,0]\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to -\infty }}\gamma (s)=p^{-}\},\\{\mathcal {M}}(L,p^{+})&=\{\gamma :&[0,\infty )\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to +\infty }}\gamma (s)=p^{+}\},\\{\mathcal {M}}(p^{-},p^{+})&=\{\gamma :&\mathbb{R} \;\;\;\to L&\,|\,{\dot \gamma }=-\nabla f(\gamma ),\lim _{{s\to \pm \infty }}\gamma (s)=p^{\pm }\}/\mathbb{R} .\end{alignedat}}

can - under an additional technical assumption specified in [1] - be compactified to smooth manifolds with boundary and corners {\mathcal {M}}(\cdot ,\cdot ). These compactifications are constructed such that the codimension-1 strata of the boundary are given by single breaking at a critical point (except in the first case we have to add one copy of L to represent trajectories of length 0),

\textstyle \partial ^{1}\overline {\mathcal {M}}(\cdot ,\cdot )\;=\;{\bigl (}\;L\cup \;{\bigr )}\;\bigsqcup _{{q\in {\text{Crit}}(f)}}\overline {\mathcal {M}}(\cdot ,q)\times \overline {\mathcal {M}}(q,\cdot ).

TODO:

  • Introduce smooth evaluation maps {{\rm {ev}}}^{\pm }:\overline {\mathcal {M}}(\cdot ,\cdot )\to L,
  • Define the length \ell :\overline {\mathcal {M}}(L,L)\to [0,\infty ] by \ell (\gamma )=a for \gamma :[0,a]\to L and \ell (\underline {\gamma })=\infty for all generalized (broken) Morse trajectories \underline {\gamma }
  • discuss boundary&corner stratification, in particular note that L\subset \partial ^{1}\overline {\mathcal {M}}(L,L) (the set of trajectories with \ell =0) is isolated from all other boundary strata (made up of generalized trajectories with \ell =\infty )
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