Timeline for The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex.
Current License: CC BY-SA 2.5
4 events
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| Aug 30, 2010 at 1:54 | comment | added | Tom Goodwillie | Suppose M and M′ are closed manifolds, h-cobordant but not simple homotopy equivalent, and of not too small dimension. An h-cobordism between them can be put inside M×ϵ, making M′ arbitrarily close to M in your sense. It seems to me that in this way you can produce a continuous path in your Hausdorff metric space from M to M'. But I don′t want $M$ and $M'$ to be in the same component. | |
| Aug 27, 2010 at 21:11 | comment | added | Dev Sinha | Yes, I think this story is what I am interested in. To follow up on your first comments, it seems to me that the Hausdorff metric restricts to the standard topology on the space of subspaces homeo to a given (compact) one (Emb(X, R^\infty)/Homeo(X)) - am I wrong on that? Though that doesn't preclude it from giving something funny when we are just restricting the homotopy type of the subspace. I wonder what happens when you think of submanifolds (of a fixed dimension?) homotopy equivalent to a given one. | |
| Aug 26, 2010 at 21:06 | vote | accept | Dev Sinha | ||
| Aug 26, 2010 at 12:57 | history | answered | Tom Goodwillie | CC BY-SA 2.5 |