alternative lecture notes - Rational points and algebraic cycles

quadrant spectral sequence
We care only about the terms with t ≥ s.
E 2 st := Hs (B, π t (F )) =⇒ π t−s Sec(E → B).
Another way to think about sections of X → Spec k (for perfect k): they are elements of
X(k) Γ k .
Let X be a space with an action of a group G. We would like to understand the homotopy
groups of X G . This is impossible, because X G is not invariant up to homotopy, as the
following example shows:
Compare X = R with translation action of Z with a point with trivial action of Z. Here
X G is empty in the first case, and a point in the second case.
Solution: define homotopy fixed points.
A fixed point is the same as a G-equivariant map from a point (with trivial G-action) to
X. We should be allowed to replace the point with a contractible space with an action of
G. When are two such maps to be considered the same? To explain this, use the universal
contractible space EG with G-action: for any contractible space C with G-action, there is
a G-equivariant map EG → C, unique up to homotopy. The space EG is unique up to
homotopy. Another equivalent definition: EG is a contractible space with free G-action.
(Free means that for every x ∈ X there exists a neighborhood U of x such that gU ∩ U for
all g ≠ 1 in G.)
Example 2.1. EZ = R with translation action. E(Z × Z) = R 2 with translation action.
Example 2.2. S ∞ := ⋃ S n ⊆ R ∞ is E(Z/2Z): the nontrivial element acts as the antipodal
map.
Define X hG := Map G (EG, X), with the compact-open topology. Choosing a different EG
gives a homotopic space X hG . This is functorial in X.
Properties:
(1) There is a natural map X G → X hG .
(2) Let C be a contractible space with a (not necessarily free) G-action. Every G-map
C → X defines an element of π 0 (X hG ).
(3) If f : X → Y is a weak equivalence (i.e., isomorphism on all homotopy groups), then
f hG : X hG → Y hG is a weak equivalence. There is a second quadrant spectral sequence
E 2 st := Hs (G, π t (X)) =⇒ π t−s (X hG ).
We have an obstruction to the existence of a homotopy fixed point:
H 2 (G, π 1 (X)), H 3 (G, π 2 (X)), . . . .
This is a special case of what we did before: define the Borel construction
(X × EG)/G → EG/G =: BG = K(G, 1).
Here EG is the universal cover of BG. We have π 1 (BG) = G and π n (BG) = 0 for n > 1.
Also, H n (G, A) = H n (BG, A).
Now (X × EG)/G → BG is a fibration with fiber X. Sections are the same as homotopy
fixed points.
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