so you'd expect an equivariant multi-jet, which is not

and actually the space of all multi-jets is Not defined simply by taking direct sums?

ANYWAY, the thing is, this bothers me because \(S\) is a finite set, so it is invariant under the action given by the symmeric group right?

anyway for multi-germs you set the multi-jet, which is defined as the tuple of mono-germs at the points

for example, if \(S=\{x_1,x_2\}\), we have

𝑗ᵏ₂𝑓=(𝑗ᵏ𝑓(𝑥₁),𝑗ᵏ𝑓(𝑥₂))

anyway then you have multi-germs, which is when you take the germ for a finite set \(S\) instead of a point (you can think of it as having a bundle of mono-germs, since manifolds are usually assumed to be Hausdorff because life is too short to not assume everything is Hausdorff)

(See Golubitski and Guillemin's 1973 "Stable Mappings and Their Singularities" for details -- I think it's Chapter II, Section 2, Jet bundles)

so when you have a germ of smooth mapping between manifolds \(f\colon (N,x) \to P\) you define the \(k\)-jet of \(f\) at \(x\) as the class of all \((N,x) \to P\) that have the same Taylor series expansion as \(f\) up to order \(k\), right?

(Taylor expansion can be computed taking charts in the source and target; while the coefficients are tied to the choice of charts, it can be proved that "having the same coefficients" is a consistent definition so long as you fix the charts)

actually there's a lot to be asid about "no the idea is simple and straightforward but the calculations took me a whole page"

is there a name for having a mapping \(f: N -> W\) and "lifting it" on both source and target? so like, a name for getting F from f in this diagram: https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRABUA5EAX1PUy58hFAEZyVWoxZt2AdV78QGbHgJEyoyfWatEIbnwGrhRcVuo6Z+hT0kwoAc3hFQAMwBOEALZIyIHAgkcSldNgAxRXcvX0R-QKQAJktpPRAAHXS0LCiQTx9g6gTEAGYUsP1M7IByXPzY5ICg0uoGOgAjGAYABUE1ERAGGDccEHLrPN4KHiA

for the curious; yes, those T stand for tangent bundles

2-page proof, almost 3-page actually

a prof during the master's told us you can tell something is a theorem when the proof is long as hecc

and honestly, if it wasn't just two statements stapled together (because they're matching statements for you and your bestie) i would say that and call it a theorem lol

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Singularity theory PhD student, wading the coboundary between algebraic and differential geometry.

I come here very occasionally and speak about maths in a very laid-back and offhand way, so this is probably not the place you want to be for your daily dose of deep insights in geometry.

Joined Oct 2018