The sphere spectrum
BID, Lisboa 2017
  1. Extensions of number systems
2. Negative numbers
3. Briefly on sheep
4. A trade deficit
5. Out in space
6. Ulla and Henriette live in RP
7. Negative sets
8. The sphere spectrum
9. Brave New World

Ulla and Henriette live in RP

Summarizing, the space Σ is a disjoint union of components - one component for each natural number (corresponding to the number of elements there might be in a finite set).

to veier som er like However, each component is in itself a fancy space (the ones corresponding to 0 and 1 are admittedly not that fancy, but from 2 on they become increasingly fascinating).

Take the component where {Ulla, Henriette} lives. There we have non-trivial path

f{Ulla, Henriette} -> {Ulla, Henriette}
given by cofusing Ulla and Henriette. This path is a loop, and it is not contractible (without letting go of the endpoints). On the other hand; if we use f twice we haven't done anything, so we insert a 2-cell so that cofusing twice is homotopic to not confusing anything.

Remember that in the picture, after identification there is just one vertex {Ulla, Henriette} and one edge f: the equality can have length zero.

So, {Ulla,Henriette} live in RP2. However, it does not stop there; we have

fff = f,
which - when you draw it - shows that {Ulla,Henriette} live in RP3 as well... and this continue to infinity:

{Ulla, Henriette} live in RP

RP is just another word for the classifying space2 of the group Σ2 with two elements.

Σ = 0 + BΣ1 + BΣ2 + BΣ3 + BΣ4 + ...

Negative sets
Bjørn Ian Dundas
2017-07-21 14:40:56 UTC