Timeline for Computing K-theory for cellular vector bundles
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 9, 2019 at 14:58 | answer | added | Mark Grant | timeline score: 2 | |
| Dec 9, 2019 at 13:08 | comment | added | Lennart Meier | @ViditNanda This might be true, but two words of "warning": 1) Going to the colimit $BGL(\mathbb{C})$ won't represent a cohomology theory. (For this you would additionally have to take a plus construction and obtain algebraic K-theory of $\mathbb{C}$.) 2) You usually won't get anything "finitely generated". For example $[S^1, BGL_1(\mathbb{C})] \cong GL_1(\mathbb{C})$. | |
| Dec 9, 2019 at 12:30 | comment | added | Vidit Nanda | @LennartMeier Sure, eventually I'd like to use a more complicated target category than the 1-truncation, but my impression is that the flat connection/local system case might be easier as a starting point. | |
| Dec 9, 2019 at 8:17 | comment | added | Lennart Meier | The "problem" with the definition is easily explained: You consider $GL_n(\mathbb{C})$ as a discrete group. It is also true in topological spaces that every map $X \to BGL_n(\mathbb{C})$ (where $GL_n(\mathbb{C})$ viewed with the discrete topology) factors homotopically over the one-truncation, which agrees with $B\pi_1(X)$ if $X$ is path-connected. This way, you only get vector bundles with flat connection. If you want all, you have to map your face-poset/simplicial set into the singular complex of the topological $BGL_n(\mathbb{C})$ or any Kan complex equivalent to that. | |
| Dec 8, 2019 at 19:51 | comment | added | John Wiltshire-Gordon | Yes, I think so, since the face poset geometrically realizes to a subdivision of $X$. | |
| Dec 8, 2019 at 19:00 | comment | added | Vidit Nanda | @JohnWiltshire-Gordon with $G = \pi_1(X)$, right? | |
| Dec 8, 2019 at 17:53 | comment | added | John Wiltshire-Gordon | The face relations act invertibly on one of your vector bundles, so this action factors through a localized category in which all these arrows have inverses. This new category is the groupoidificaiton of the face poset. If $X$ is connected, then this groupoid is connected, and hence equivalent to some group $G$. A vector bundle in your sense is then the same as a complex representation of $G$. | |
| Dec 8, 2019 at 16:08 | history | asked | Vidit Nanda | CC BY-SA 4.0 |