group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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).
due to Hopf 35, which is compatible with the inclusions as $n$ varies
and hence induces an inclusion
check
Composing this with the J-homomorphism gives a map
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 $X$
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 $\mathbb{R}P^\infty$).
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.):
Notice that Snaith's theorem asserts that this map becomes in fact an isomorphism to reduced K-theory after quotienting out the Bott generator $\beta \in \Sigma^\infty \mathbb{C}P^\infty_+$.
The original formulation is due to
A strengthening was obtained in
Review is in
The analogous statement for complex projective space and complex topological K-theory is due to
Last revised on September 10, 2018 at 08:16:02. See the history of this page for a list of all contributions to it.