where the generator [1]∈ℤ/24[1] \in \mathbb{Z}/24 is represented by the quaternionic Hopf fibration S 7⟶h ℍS 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4.

Under the Pontrjagin-Thom isomorphism, identifying the stable homotopy groups of spheres with the bordism ring Ω • fr\Omega^{fr}_\bullet of stably framed manifolds (see at MFr), this generator is represented by the 3-sphere (with its left-invariant framing induced from the identification with the Lie group SU(2) ≃\simeq Sp(1) )

π 3 s ≃ Ω 3 fr [h ℍ] ↔ [S 3]. \array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. }

Moreover, the relation 24[S 3]≃02 4 [S^3] \,\simeq\, 0 is represented by the complement of 24 open balls inside the K3-manifold (MO:a/44885/381, MO:a/218053/381).

Kahn-Priddy theorem

The Kahn-Priddy theorem characterizes a comparison map between stable cohomotopy and cohomology with coefficients in the infinite real projective space ℝP ∞≃Bℤ/2\mathbb{R}P^\infty \simeq B \mathbb{Z}/2.

Boardman homomorphisms

To ordinary cohomology

Consider the unit morphism

𝕊⟶Hℤ \mathbb{S} \longrightarrow H \mathbb{Z}

from the sphere spectrum to the Eilenberg-MacLane spectrum of the integers. For any topological space/spectrum postcomposition with this morphism induces Boardman homomorphisms of cohomology groups (in fact of commutative rings)

(1)b n:π n(X)⟶H n(X,ℤ) b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z})

from the stable cohomotopy of XX in degree nn to its ordinary cohomology in degree nn.

(Arlettaz 04, theorem 1.2)

To topological modular forms

Write 𝕊\mathbb{S} for the sphere spectrum and tmf for the connective spectrum of topological modular forms.

Since tmf is an E-∞ring spectrum, there is an essentially unique homomorphism of E-∞ring spectra

𝕊⟶e tmftmf. \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,.

Regarded as a morphism of generalized homology-theories, this is called the Hurewicz homomorphism, or rather the Boardman homomorphism for tmftmf

Proposition

(Boardman homomorphism in tmftmf is 6-connected)

The Boardman homomorphism in tmf

𝕊⟶e tmftmf \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf

induces an isomorphism on stable homotopy groups (hence from the stable homotopy groups of spheres to the stable homotopy groups of tmf), up to degree 6:

π •≤6(𝕊)⟶≃π •≤6(e tmf)π •≤6(tmf). \pi_{\bullet \leq 6}(\mathbb{S}) \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow} \pi_{\bullet\leq 6}(tmf) \,.

(Hopkins 02, Prop. 4.6, DFHH 14, Ch. 13)

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory\;M(B,f) (B-bordism):

relative bordism theories:

MOFr, MUFr, MSUFr

equivariant bordism theory:

global equivariant bordism theory:

algebraic:

algebraic cobordism

References

The concept of stable Cohomotopy as such:

Frank Adams, part III, section 6, p. 204 of: Stable homotopy and generalised homology, 1974
John Rognes, p. 1 of: The sphere spectrum, 2004 (pdf)

Discussion of stable Cohomotopy as framed cobordism cohomology theory:

Robert Stong, Chapter IV, Example 1, p. 40 of Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016)

Discussion of stable Cohomotopy of Lie groups:

C. T. Stretch, Stable cohomotopy and cobordism of abelian groups, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981, pp. 273-278 (doi:10.1017/S0305004100058734)
Ken-ichi Maruyama, ee-invariants on the stable cohomotopy groups of Lie groups, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 (euclid:ojm/1200780982)
Sławomir Nowak, Stable cohomotopy groups of compact spaces, Fundamenta Mathematicae 180 (2003), 99-137 (doi:10.4064/fm180-2-1)

The identification of stable cohomotopy with the K-theory of the permutative category of finite sets is due to

Michael Barratt, Stewart Priddy, On the homology of non-connected monoids and their associated groups, Commentarii Mathematici Helvetici, 47 1 (1972) 1–14 [doi:10.1007/BF02566785, eudml:139496]
Graeme Segal, Categories and cohomology theories, Topology vol 13, pp. 293-312, 1974 (doi:10.1016/0040-9383(74)90022-6, pdf)