User pablo - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T22:08:05Z http://mathoverflow.net/feeds/user/30555 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118574/conley-index-for-isolated-invariant-sets-with-no-exit-points Conley index for isolated invariant sets with no exit points Pablo 2013-01-10T21:17:27Z 2013-01-11T12:49:57Z <p>Conlay described in $\textit{Isolated Invariant Sets and the Morse Index (1976)}$ the bases of what would be known as Conley Index Theory. </p> <p>For the sake of simplicity let's think of vector fields defined on manifolds (a more general situation is enough by just considering locally compact spaces). Im going to pose a few basic notions needed for the question so even someone who is not familiar with the topic may give an answer.</p> <blockquote> <p>A set (in the phase space) is called invariant if it is the union of solution curves. It is isolated if it is the maximal invariant set in some neighborhood of itself. A compact such neighborhood is called an isolating neighborhood for the invariant set.</p> <p>An isolating neighborhood is an isolating block if the integral curves through each boundary point of the neighborhood goes immediately out of it in one or the other time direction.</p> </blockquote> <p>And finally:</p> <blockquote> <p>The $\textit{(Conley)}$ index is the homotopy type of the pointed space obtained from a block on collapsing the set of exit points (points in the boundary where integral curves go out the neighborhood) to one point.</p> </blockquote> <p>Well, I've read that when we have an isolating block, say $N$, that has no exit points, we would have to collapse the empty set $\emptyset$ to one point (?), so there's a convention that the resulting space is $$(N \bigcup \lbrace \star \rbrace, \star)$$ that is, the disjoint union of the space $N$ and an external point, all of it pointed at that external point).</p> <p>So the questions are</p> <ul> <li>Why this convention makes sense (apart from that $\textit{it just works}$)? Is that the natural way of defining the operation "collapse to a point'' when all you have to collapse is $\textit{nothing}$ (i.e. $\emptyset$)?</li> <li>Is there a bigger frame where we don't need this convention because a more general rule contains it as a particular case?</li> </ul> http://mathoverflow.net/questions/118574/conley-index-for-isolated-invariant-sets-with-no-exit-points/118610#118610 Comment by Pablo Pablo 2013-01-12T10:37:30Z 2013-01-12T10:37:30Z Thank you, that was pretty much what I was looking for. http://mathoverflow.net/questions/118574/conley-index-for-isolated-invariant-sets-with-no-exit-points/118591#118591 Comment by Pablo Pablo 2013-01-10T23:52:32Z 2013-01-10T23:52:32Z Well, I knew that. I don't know if my questions doesn't have an answer but what I was looking for is for the reasaon of the operation of ''collapsing the emptyset to one point'' or, stating it differently, if that is the way it is extended in other cases.