Timeline for Examples of topologically non trivial complete submanifolds in infinite dimensional Banach spaces
Current License: CC BY-SA 4.0
14 events
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| Aug 23, 2023 at 14:58 | comment | added | 0x11111 | @TomGoodwillie OK. Taking $X=T^\infty= \lim T^n$, we get $\pi(X)=Z^\omega$ ( $\omega$ - first infinite ordinal(, $H_n(X;Z)=Z^{o_n}$, where $o_n$ is the ordinal type of the set of trees of height n on $\omega$. Is there something like "Banach homology" where we allow explicitly infinite dimensional faces? ( i.e., we promote CW to "$\omega$ CW"). And where $H_nX$ are TVS. | |
| Aug 23, 2023 at 14:57 | comment | added | 0x11111 | @PietroMajer Is there something like "quantitative Frechet homotopy" theory, that addresses questions like "What is the strongest topology in which two homeomorphic sets are homotopy equivalent by a homotopy of given class?" and then develops the algebraic machinery ($\pi_nX$ etc.) for these Frechet maps? | |
| Aug 23, 2023 at 14:56 | history | edited | 0x11111 | CC BY-SA 4.0 |
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| Aug 2, 2023 at 14:04 | comment | added | Pietro Majer | @Ox11111 For instance the unit sphere of any Banach space is not contractible by means homotopies of the form id+compact, because of the Schauder degree, just as in finite dimension. | |
| Aug 2, 2023 at 13:59 | comment | added | Tom Goodwillie | @0x11111 In the embedding that I am thinking of, there is a projection on a 2-dimensional vector subspace such that the image of the infinite-dimensional torus is a circle. So the image of the closure is also that circle, and the closure of the torus has a circle as a retract, therefore is not contractible. | |
| Aug 2, 2023 at 13:47 | comment | added | 0x11111 | @BillJohnson Thanks. Is there analogy of topologically nontrivial algebraic subgroups of automorphisms of non reflexive spaces, e.g. C[0,1]? | |
| Aug 2, 2023 at 13:43 | comment | added | 0x11111 | @PietroMajer What are the examples you have in mind? | |
| Aug 2, 2023 at 13:43 | comment | added | 0x11111 | @TomGoodwillie How do you prove noncontractibility of the torus that you are thinking about? | |
| Jul 20, 2023 at 12:58 | comment | added | Bill Johnson | For contractibility of the general linear group you might look at Mitiagin's classical paper The homotopy structure of the linear group of a Banach space mathnet.ru/php/… | |
| Jul 20, 2023 at 6:56 | comment | added | Pietro Majer | As a side remark, note that in an infinite dimensional setting there are several different categories of continuous maps (identity+compact, Fredholm, etc) which may make a topologically trivial object non-trivial | |
| Jul 16, 2023 at 15:48 | comment | added | James E Hanson | I just realized I dropped the word 'more' from my comment. I meant to say 'are you looking for more structure.' | |
| Jul 16, 2023 at 14:47 | comment | added | Tom Goodwillie | What do you mean by "After completion, they become contractible, as is the case with the above torus"? If your infinite-dimensional torus is topologized as a product of circles, then it is compact. If not, then it is still true that the first embedding that I think of has non-contractible closure. | |
| Jul 16, 2023 at 14:36 | comment | added | James E Hanson | There are general methods to isometrically embed metric spaces into certain Banach spaces. Would this answer your first question, or are you looking for structure than this? | |
| Jul 16, 2023 at 12:29 | history | asked | 0x11111 | CC BY-SA 4.0 |