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Apr
23 |
revised |
Repeated Homotopy Category of Chain Complexes
added 192 characters in body |
Apr
23 |
asked | Repeated Homotopy Category of Chain Complexes |
Jun
28 |
awarded | Scholar |
Jun
28 |
accepted | Are evaluation maps for sections of a fiber bundle weak homotopy equivalences? |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
I thank you for the help and clarification. To sum up: the statement in my first paragraph were true if I had included contractibility of the base space, which of course makes for much more sensible statement. I will now mark your answer as accepted, unless I have misunderstood your last comment. |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
@Torsten Thank you for convincing me of the falseness of the statement in my first paragraph in general. |
Jun
28 |
revised |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
update |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
Thus my concrete current question is: why is the evaluation map $\varepsilon:\Gamma^0(E^k_{\omega})\to (p^k_{\omega})^{-1}(0)$ a w.h.e., noting that $(p^k_{\omega})^{-1}(0)\subset J^k_0(D^m,F)$? |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
To me it also seemed to be a more general phenomenon which is why I formulated the more general variant (which may be too general). If I knew that the evaluation map is a w.h.e. I'd be indeed done, this is why I was hoping to get a positive answer to the first question. You have shown that in the first question the space of sections and the fiber over the point are homotopy equivalent, but I would need the fact that explicitly the evaluation maps are appropriate (weak) homotopy equivalences, which by the commend of Torsten and my intuition seems to be untrue in general. |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
I forgot to type that $J^k_0(D^m,F)$ is supposed to be the space of $k$-jets "at $0$" of smooth maps $D^m\to F$. |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
@Dan I forgot to include your name for you to receive the notification. |
Jun
28 |
comment |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
Unfortunately I'm not sure I understand the argument completely. I know that $E^k_{\omega}(0)$ is a subspace of the space $J^k_0(D^m,F)$ of $k$-jets of smooth maps $D^m\to F$ where $F$ is the fiber of $E$. You are arguing that thus $\Gamma^0(E^k_{\omega})$ is homotopy equivalent to $(p^k_{\omega})^{-1}(0)$ which we know is homeomorphic to $\Gamma^0(E^k_{\omega}(0))$. My gripe is that this does not show that the explicit map $\rho$ is a w.h.e. but only that the domain and range are (weak) homotopy equivalent. It's very well possible that I'm missing the point here. |
Jun
28 |
awarded | Student |
Jun
28 |
revised |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
corrections |
Jun
28 |
awarded | Editor |
Jun
28 |
revised |
Are evaluation maps for sections of a fiber bundle weak homotopy equivalences?
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Jun
28 |
asked | Are evaluation maps for sections of a fiber bundle weak homotopy equivalences? |