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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 8th 2021

As is well known, it is our manifest destiny as 21st century algebraic topologists to compute homotopy groups of spheres. (Wilson 13, p. 1)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 8th 2021

For the last 50 years one of the basic problems in algebraic topology has been the determination of the homotopy groups of spheres (Mahowald-Ravenel 87, Sec. 1)

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeFeb 9th 2021

It is all too evident I am not an expert on homotopy theory, and the books I am bold enough to write now on foundational matters are very likely to be looked at as “rubbish”, too, by most experts, unless I show up with $\pi_{147}(S^23)$ as a by-product (whereas it is for the least doubtful I will\ldots)

(Lettre d’Alexandre Grothendieck à Ronald Brown, 06.09.1983, In: Correspondence Alexandre Grothendick – Ronald Brown.)

As a counterpoint to the view expressed there?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 9th 2021

Your quote would nicely fit into the entries Tremolo of the Test Categories or Decrescendo of the Derivators .

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 9th 2021

Maybe an interesting question to ponder here is the $\infty$-categorified version of ultrafinitism:

Where ultrafinitism wonders whether extremely large elements of the naive integers $\mathbb{Z}$ may become so intangible in practice as to be irrelevant in theory, the homotopy-ultrafinitist, envisioned hereby, also tends to doubt the relevance of elements of the true integers $\mathbb{S}_\bullet$ once their degree becomes very large:

From which degree on is any practical impact of the concrete knowledge of the stable stem in that degree so inconceivable as to render its concrete computation more of an embarrassment than an achievement, a mathematical indecency whose obscenity only grows with the effort that went into it?

Indeed, cutting edge developments still struggle with providing practical meaning even for the very lowest stems, just think of Kapranov’s speculation (here) which conjured a grand vision only to extend established meaning from the second to the third stem, and tentatively so.

Now, since Kapronov appeals to string physics for this, Hypothesis H might come to mind, where the third stem leads a robust existence of the M-brane charge group in black M2 backgrounds. But isn’t the interpretive power of Hypothesis H also restricted to low lying stems, in degrees $\leq 11 - 4$, at best?

But this gets interesting as one recalls that the 11 dimensions here are just the classical ones of a super-exceptional geometry whose bosonic body is of dimension no less than… 528. Since we know for a fact that the flux 4-form generally roams in the full 528-dimensional exceptional spacetime, a super-exceptional Hypothesis H would plausibly need to be speaking of the compactly supported 4-Cohomotopy of Euclidean spaces of dimension up to 528.

This would, at a stroke, provide practical (physical) meaning to the $n$-stems in degrees $\leq 524$. Still finite, but much larger than 3.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeFeb 9th 2021

Just devise a homotopic RSA-like encryption system using the true integers and the there’ll be no limit.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 9th 2021

Incidentally, it finally dawns on me what the tone of Adams’s comment on the homotopy groups of spheres (here) reminds me of: This is Melville’s, speaking about the Razor Back Whale (here).

With all due respect to respect to anyone interested in them, they are a mess.

Melville:

Let them go. I know little more of them, nor does anybody else.

Chorus:

Let them go, they are a mess.

Curtain.