Talk:Homotopy theory

I tagged the use of the vague word "nowadays", and @TakuyaMurata reverted it, commenting, "I don’t think we need a specific date here." Rather than relying on any one editor's opinion, I am seeking consensus. I don't see any downside to making this more specific, but could not easily nail down the time period. ~TPW 15:40, 31 January 2023 (UTC)

I don’t think it’s easy to find out a specific time period. Also, I don’t think the specificity is particularly helpful here. Is there any reason for wanting a specific time? —- Taku (talk) 19:10, 5 February 2023 (UTC)

I wonder if this could be simplified by using Exp(LieAlgebraContaining(f(x)*d/dx))*(n_Sphere->x) and getting the (possibly disconnected) components. What is the actual formula for homotopy? I'll try (GroupExp(LieAlgebraContaining(f(x)*d/dx)))*(n_Sphere->x), but then what does one do? For the first homotopy class (the fundamental group) one might want the CayleyTable(e^(2*i*pi*f^(-1)(y)*d/df^(-1)(y))), but how does one get the homotopy classes? Should one do: CayleyTable(e^(2*i*pi*d/d(GroupExp(g(y)*d/dy)*ln(f^(-1)(y)))))?? Now would that work? Can one generalize this U(1)-symmetry method to SU(n)?? What happens? Why does one use a sphere instead? So one seems to be classifying f^(-1)(y) by deformation? What about the subspaces of y? Should one care? One can use the complex exponential instead of U(1) to make the U(1) handle a subspace instead, right? Wouldn't one get: CayleyTable(e^(2*(SU(n)+s)*pi*d/d(GroupExp(g(y)*d/dy)^(inf)*ln(f^(-1)(y))))), via abuse of notation? Now why would there be only 1 way to 'multiply'. It seems a group (perhaps a rank 2 representation) might not work. Lie 3-algebras? Lie 1000-algebras? Something isn't working. I guess one could have a 2d representation by picking out SU(n) generators, but why? Given CayleyTable(e^(2*(SU(n)+s)*pi*d/d(GroupExp(g(y)*d/dy)^(inf)*ln(f^(-1)(y))))), this seems to describe homotopy groups but with SU(n) instead of a sphere? Basepoint is there, check! Homotopy seems to be there (the infinite deformation time-dilation possible abuse of notation), check! Number n is there, check! Groups are there, check! Am I missing anything? Should I use: CayleyTable(e^(2*(SU(n)+s)*pi*d/d(GroupExp(g(y)*d/dy)^(inf_{nonvisiciousbloodymess})*ln(f^(-1)(y))))) to denote the equivalence class more clearly (but a bit morbidly)? I have synesthesia so the 'nonvisiciousbloodymess' subscript helps me see it better. ~2026-29937-4 (talk) 14:36, 24 February 2026 (UTC)

Basically, the binary code for that subscript is Do the papers bleed pen ink: Yes, It is viscous: no (or not likely), more like water. Stick the papers together and the ink 'runs' like a cup of water, that is the best way to explain my bad-sounding notation. If it was 'viscous' then it might not 'look like' a homotopy in my mind, there would be 'clots' (of ink) which obstruct the homotopy in this analogy, which might happen in other theories that are currently under my construction. Here it is like pen, paper, and water. ~2026-29937-4 (talk) 15:25, 24 February 2026 (UTC)

These clots could be some kind of 'time machine'? I'd have to figure that out someday. ~2026-29937-4 (talk) 15:27, 24 February 2026 (UTC)

Complex 4th power defect could be a clot in a homotopy (the paper-water), but it is also a time machine (it can go way back in time because time and space get swapped). Going near x^(1/4) flings you backward. Intrinsically with no surroundings it is not a time machine, but if the surrounding space is like y=x, then crazy things happen, which one could consider the clot to be like that in an extended theory. If only a human or dog could fit inside one of these possible defects things get interesting. Special and general relativity don't like them, but they could be possible by removing some assumptions about what space should be. ~2026-29937-4 (talk) 15:31, 24 February 2026 (UTC)

Is there such thing as a pseudo-homotopy that 'lies to us' in a specific way to obtain a larger equivalence class? Same question with topology. Same question with curvature. What is beyond topology? What is space, really? Can space be cut like topologists suggests, the evidence shows that contrary, given quantum entanglement. Does a single atom know everything, since background independence implies the inability to forget, since the notion of gravity is path-dependence? If one can't forget, then where are all the memories of the future, let alone the past? The equivalence class of topology might be too small and describe the 'vacuum' (in gravity connotation, more empty). Could a larger equivalence class describe matter? ~2026-29937-4 (talk) 01:19, 24 February 2026 (UTC)

Do you mean to say: can we consider an equivalence relation other than homotopy? That’s certainly true and that’s something studied in abstract homotopy theory. As for a space, we can consider space-like objects like pro-spaces in shape theory. —- Taku (talk) 13:41, 24 February 2026 (UTC)

I maybe figured this all out. Couldn't one represent homotopy as a lie group of subspaces, where one creates an 'equivalent circuit' in some sense. The fundamental group would be the 'circuits' with U(1), and other homotopy groups would then attach higher dimensional objects to replicate the space in question? It seems like homotopy finds a group structure to sets of possible subspaces if I am right. The torus immediately becomes it's lie group, since it is. Other spaces are trickier. How about cubes? In which way does one attach them? All possible sides? One is ending up with a multiplication with more than 2 arguments? It would be written like A(B,C) for cubes? How would that be a group with only 2 arguments? Does one consider AB and AC?? That makes a loop, but isn't that only the fundamental group? It says one joins the spheres at the equator (by connecting the poles or something), or something, which is a revolved loop. Does one revolve the loop? One could consider the cayley table of loops for the fundamental group. What next? A higher dimensional hole, right? That should then mean there exists an object not contractible to point, right? If that is true, then one has a revolved loop to fill the object? Technically the object would be the hole, right? So a revolved loop would likely fill it (somewhat locally), right? How much of this would be enough? An object homotopic to the n-sphere? Could one consider that as a plane that is not contractible in every direction? If there is a n-dimensional hole, then there would be repulsion applied to some plane (somewhat locally). What is interesting is that the sphere is 'loops everywhere'. At every point on the sphere (at least for 2-spheres) there is a loop and a point. So it is like a ball of yarn. So would that mean that one makes it as loopy as possible for the hole under consideration? If a 2-sphere hole existed, then the 2-sphere around it couldn't be contracted, right? So there is a 2 dimensional region that cannot be contracted to a point in any other way than itself. Is that why spheres are used? Going through any of these holes in some way is 'impossible' (it is possible, but in some sense isn't for something), since a loop cannot be 'brought' over in that way completely, so that would create another homotopy class, correct? So if one searches for 'wormholes' one could call this a 'wormhole group', via the composition of 'wormholes'. One then considers the mouths to be 'everywhere' making sense with the whole homotopy thing, since wormholes are made up of space. The question is, for certain complex functions have holes but they don't go anywhere else, e.g. the complex plot x^(1/4), or maybe x^4 (depending on what one favors poles/zeros), so what happens there? Does one wrap around it? The fundamental group (I guess) is the cayley table of the loops, which I know how to do. That might be non-trivial. But what about higher dimensional holes? Does one look for higher dimensional wormholes with some kind of curvature? But a cubical wormhole doesn't have much curvature? Does one look for 'cubes' in the space, rather than path-objects (which create squares)? How many points to 'loop around' would be considered? If one considers preserving location, but recovering measurement (Meas->LOC->Meas), the cayley table of operators works for the fundamental group (I think). Does one consider (Meas->LOC->Meas) with something different? On a sphere the symmetries aren't as neat. Considering two endpoints leads to an infinite explosion of values (too many mirrors, too small mirrors). Does one consider that as a single super-mirror of some kind? How does one classify it? Any 2 dimensional sphere as a hole is like any similar 2 dimensional hole that is closed on itself like that of a cube? But the 'symmetries' would be different (the fundamental group). Would one consider some kind of qubit as the information regarding the symmetry? U(1) then SU(2) then SU(3) and so on? Then the state of qubits and qutrits and their composition could be linked to homotopy theory? Compose two SU(2)s to 'multiply' the quantum state matrix and the structure of the multipliers (like on a computer chip) would be the homotopy theory? ~2026-29937-4 (talk) 22:17, 25 February 2026 (UTC)

Basically homotopy theory would decompose spaces into a lie group of interrupted lie groups, if I am right. ~2026-29937-4 (talk) 22:21, 25 February 2026 (UTC)

Can't one then make a sequence out of this. Lie groups would be 1 iteration. Homotopy theory would be the 2nd iteration. That then makes more lie groups. The 3rd iteration might recover extended space-specific homotopy theories that compose or something (a lie group of lie groups), but the discrete nature of the interruptions might not have a notion of interruption, so it appears that homotopy theory terminates the sequence? ~2026-29937-4 (talk) 22:26, 25 February 2026 (UTC)

You can read the article, or go by what I will do personally to generalize this a bit (deformation of interruptions (thus deformation of the lie groups which remains a lie group in some way) leads to recovered continuity as expected). ~2026-29937-4 (talk) 22:28, 25 February 2026 (UTC)

The question is, does homotopy theory specify the amount of lie groups needed (or 'created') for travelling a certain amount (might not even be defined) through a certain number of holes? More like how do these lie groups travel through the holes (which they can't without breaking), right? Then how does that form a set of lie groups to describe that? One then asks for a lie group describing composition of interruptions given a collection of lie groups of a certain dimension travelling through (maybe it's own) wormhole, where on the interruption one uses matrix multiplication? Interesting parallel with canonical quantization and quantum operators, right? This seems to be why BRST quantization is a thing, right? If only the formulas were explicit in that article. ~2026-29937-4 (talk) 22:35, 25 February 2026 (UTC)

Doesn't this mean that a generalization homotopy theory might not be described by the dimensionality alone? What about inserting any allowed/disallowed combination of a collection of spaces that one specifies? Would homotopy theory produce the same results for all of these spaces? Seems ridiculous but possible. If that is not possible to get the results to agree, then what next? Why spheres? ~2026-29937-4 (talk) 22:43, 25 February 2026 (UTC)

I figured out my own formula.

It is:

Mf1x=f2*G_{mult,n}(exp_G(f3_n(x))=f2*exp_G(f3_{out}(x))

Now find the multiplication tables for f3_n and f3_out for each lie group G, for a space M, forall functions (possibly bounded) f1.

This is homotopy theory on steriods.

A wide variety of topological questions basically enumerate (for the space M in the created question):

f*M*L(f2,x)=L2(f3,W_hat)(x)

Enjoy (perhaps your quantum computer from a space)!

--Misha Michael Mikhail Victor Taylor [Trophis*] ~2026-29937-4 (talk) 05:23, 7 March 2026 (UTC)

More precisely, f^(-1)*M*L(f2,x)=some_added_structure_constraints(L2(f3,W_hat)(x)) would be best. One could write that as: M*L(f2,x)=f*some_added_structure_constraints(L2(f3,W_hat)(x)). There!

For some updated homotopy theory one can do: W_hat=LieGroupExponential (via integrals and matrices and fixed points), and then use f3(x) into L2, and set the added structure to matrix multiplication. The newer homotopy theory depends on the lie group and it's dimension, rather than just the dimension if I'm right. SO(3) and SU(2) might possibly merge due to the double cover thing? ~2026-29937-4 (talk) 05:32, 7 March 2026 (UTC)

Thus topological questions like these create enumerative questions about f^(-1)*M. ~2026-29937-4 (talk) 05:38, 7 March 2026 (UTC)