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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?