Difference between revisions of "Compactified Morse trajectory spaces"

From Polyfolds.org
Jump to: navigation, search
(Created page with "Consider a smooth manifold <math>L</math> equipped with a Morse function <math>f:L\to\R</math> and a metric so that the gradient vector field <math>\nabla f</math> satisfies t...")
 
Line 17: Line 17:
 
</math>
 
</math>
 
</center>
 
</center>
 +
 +
'''TODO:''' evaluation maps

Revision as of 21:17, 20 May 2017

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: evaluation maps

Retrieved from "http://www.polyfolds.org/index.php?title=Compactified_Morse_trajectory_spaces&oldid=148"