nLab Kahn-Priddy theorem

Idea

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

Statement

Write $\mathbb{R}P^\infty \in Ho(Top)$ for the homotopy type of real projective space (an object in the classical homotopy category), and write $\Sigma^\infty \mathbb{R}P^\infty_+ \in Ho(Spectra)$ for its suspension spectrum regarded as an H-group ring spectrum in the stable homotopy category.

For each $n$ there is a canonical inclusion (see Whitehead 83, p. 20).

$$\mathbb{R}P^{n-1} \hookrightarrow O(n)^+$$

due to Hopf 35, which is compatible with the inclusions as $n$ varies

$$\array{ \mathbb{R}P^{n-1} &\hookrightarrow& O(n) \\ \downarrow && \downarrow \\ \mathbb{R}P^n &\hookrightarrow& O(n+1) }$$

and hence induces an inclusion

$$\mathbb{R}P^\infty \hookrightarrow O$$

check

Composing this with the J-homomorphism gives a map

$$\phi \;\colon\; \Sigma^\infty \mathbb{R}P^\infty \longrightarrow \mathbb{S}$$

from the H-group ring spectrum of infnite real projective space to the sphere spectrum.

Then for $X$ a connected 2-primary finite CW-complex, the function that takes stable maps into this H-group ring spectrum to maps to the sphere spectrum, hence to the stable cohomotopy of $X$

$$\phi_X \;\colon\; [\Sigma^\infty X_+ , \Sigma^\infty \mathbb{R}P^\infty_+ ] \overset{surj.}{\longrightarrow} [\Sigma^\infty X_+, \mathbb{S}]$$

is surjective.

In this form this is stated in Adams 73, lemma 3.1 (see the notation introduced below lemma 2.2: for $p = 2$ then Adams’s $L$ is $\mathbb{R}P^\infty$).

Analog for complex projective space

The statement was announced in the above form in Segal 73, prop. 2, where the analogous statement for complex projective space and topological K-theory is proven (see this prop.):

$$[\Sigma^\infty X_+ , \Sigma^\infty \mathbb{C}P^\infty_+ ] \overset{surj.}{\longrightarrow} K_{\mathbb{C}}(X) \,.$$

Notice that Snaith's theorem asserts that this map becomes in fact an isomorphism to reduced K-theory after quotienting out the Bott generator $\beta \in \Sigma^\infty \mathbb{C}P^\infty_+$.

References

The original formulation is due to

• Daniel Kahn, Stewart Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 (Euclid)

A strengthening was obtained in

• John Frank Adams, The Kahn-Priddy theorem, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematical Proceedings of the Cambridge Philosophical Society 55, 1973

Review is in

• George Whitehead, pages 20, 21 of Fifty years of homotopy, Bulletin of the AMS, Volume 8, Number 1, 1983 (pdf)

The analogous statement for complex projective space and complex topological K-theory is due to

• Graeme Segal, The stable homotopy of complex of projective space, The quarterly journal of mathematics (1973) 24 (1): 1-5. (pdf, doi:10.1093/qmath/24.1.1)