There are infinite families in the stable homotopy groups of spheres; many of these can be seen by looking for "periodicity" phenomena in the Adams-Novikov spectral sequence. An example is the image of the $J$-homomorphism. This particular infinite family can be described algebraically (at least at odd primes). If I understand correctly, these elements correspond to elements of filtration $1$ in the ANSS. That is, they are in $\mathrm{Ext}^1_{BP_*BP}(BP, BP)$ via the exact sequence of comodules $$0 \to BP \stackrel{p}{\to} BP \to BP/p \to 0.$$ The invariant elements in $BP/p$ are the powers of $v_1$, and the coboundaries of these live in $\mathrm{Ext}^1(BP, BP)$. These correspond to multiples of the image of $J$.

Other infinite families in the stable homotopy groups of spheres can be obtained from a similar procedure. One nice thing about the image of $J$, though, is that it has a geometric interpretation: if the stable homotopy groups of spheres are identified with the framed cobordism groups, then the image of $J$ corresponds to framed spheres. Are there similar geometric interpretations of these more exotic infinite families?

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    $\begingroup$ Not necessarily related to infinite families but in Hopkins ICM address he discusses what is known geometrically about the first 16 stable homotopy groups (arxiv.org/abs/math/0212397) $\endgroup$
    – Drew Heard
    Jun 29 '12 at 9:44

This has been a burning question for quite some time, but not much is known. Surely, people believe that the next layer (i.e. the $\beta$-family) should also admit a geometric description, although as of yet nobody has been able to find one.

The geometry behind the $J$-homomorphism (homotopy of the stable orthogonal group) is very closely related to the geometry that underlies $K$-theory, so I suppose the same will be true for the $\beta$-family. There's a lot of remarkable work going on from people who try to provide a geometric construction of elliptic cohomology, and you could ask if their stuff contains the germ of a geometric construction of the $\beta$'s. Maybe some of the finite dimensional models of the String group could play a role there?

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    $\begingroup$ I guess a related question (perhaps I'll ask it here in a few days) is why people expect a geometric interpretation of elliptic cohomology (I keep hearing words such as "loop groups" and "von Neumann algebras" thrown around.) $\endgroup$ Jun 29 '12 at 19:17

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