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I’m considering rewriting and submitting section 5.1 (down to Corollary 5.17) of my thesis to the nJournal. The core result is a minor extension of a special case of a theorem by Wada: certain path spaces have local connectivity properties if the original space has said property one dimension up. Apart from the main result, which was mostly a technical necessity for a construction in my thesis, the main interest would be in the collection of neighbourhood bases I give for the compact-open topology on certain path spaces. There are corollaries that certain quasitopological homotopy groups are actually discrete, but I didn’t bother with these at the time.
I would endeavour to nlabify this as much as possible, but I might need help with pictures. I have the source of the pictures (a graphics package, unfortunately), and can produce jpegs, gifs etc of them.
Any thoughts?
Hi David,
I just have a second to look into this, will try to react in more detail a bit later.
We need to finally install the editorial board! That should answer such questions.
One quick question:
Is theorem 5.12 the main statement here? Meant to improve on Wad55? (Maybe this question just means: when you submit, there needs to be an abstract :-)
Another question:
Do you really mean to include $X^I$ in the list of spaces that are $(n-1)$-connected if $X$ is $n$-connected? Seems to me that $X^I$ is weak homotopy equivalent to $X$, unless the endpoints of the paths are fixed?!
Concerning the remark above the theorem, on transgression of structures: Nikolaus and Waldorf will have something on this, soon, see here.
Yes, $X^I$ is homotopy equivalent to $X$, but you can’t use a homotopy equivalence to transfer a basis for a topology, so there is no reason to expect them to have the same local properties.
Yes, I do need an abstract, but Theorem 5.12 is as close to a main theorem as one will get. There are a number of corollaries about mapping spaces of several sorts.
And as far as editorial duties go, this is merely an informal question, asking people whether they feel this sort of thing is nJournal material. If I make a formal submission before the editorial board is finalised, and the standing comittee need to decide things, then clearly I will remove myself from all discussions.
Hi,
have another second, but not more.
no reason to expect them to have the same local properties.
Ah, I see. Since I just proved that this point is easy to miss on casual reading, maybe something to keep in mind for highlighting in the abstract.
And as far as editorial duties go, this is merely an informal question, asking people whether they feel this sort of thing is nJournal material.
I feel it is perfectly fine.
Just as a feedback: I am not an expert on the details, but if I were asked to referee this, two questions would immediately spring to my mind:
Since this is about a classical result from 1955, does the author make the point for why we need to reconsider it?
Is there some example or application that goes with this and makes us feel good about having the theorem and/or its new proof?
Possibly both of these questions find plenty of answer in later sections of your thesis. Might it make sense to submit a bigger chunk of it?
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