Download Topological K-theory: problem set 7

Topological K-theory: problem set 7
Problem 1. Show that for all n ≥ 3, πn (S3 ) ∼
= π n ( S2 ).
Problem 2. Compute K ∗ (CPn ) as an additive group (the ring structure is more
complicated).
Problem 3. A prespectrum is a sequence of spaces En with maps φn : En → ΩEn+1
(the difference with a spectrum is that we don’t assume the φn to be homotopy
equivalences).
For a prespectrum E, define a new sequence of spaces E# by
En# = hocolim Ωk En+k ,
where the maps in the hocolim are induced by the φn . Show that E# is a spectrum.
It is called the spectrification of E.
In particular, for a pointed space X, we can apply this to the sequence of spaces
En = Σn X, where the structure maps φn are given by X → ΩΣX, the adjoint of the
identity on ΣX. The resulting spectrum is called the suspension spectrum of X and is
denoted by Σ∞ X.
Problem 4. Let E be a spectrum. Show that the functor
X 7→ π0 (( E ∧ X+ )#−n )
defines a homology theory. Here X+ = X t {∗} denotes the union with a disjoint
base point, and E ∧ X is the prespectrum defined by ( E ∧ X )n = En ∧ X. Thus we
not only get a cohomology theory out of a spectrum E, but also a homology theory.