Proof. (i) In the simplicial context, homotopy groups are defined after geometric realization, which
commutes with mapping cones. So it suffices to treat the case of symmetric spectra of spaces. We show that the sequence
πkX πkf −−−−→ πkY πk(incl) −−−−−→ πkC(f) πk(proj) −−−−−→ πk(S1∧X) πk(S 1∧f) −−−−−−→ πk(S1∧Y)
is exact; when we substitute definition (4.2) of the boundary map δ, this becomes the exact sequence of part (i).
Exactness atπkY: the composite off :X−→Y and the inclusionY −→C(f) is nullhomotopic, so it induces the trivial map onπk. So it remains to show that every element in the kernel ofπk(incl) :πkY −→
πkC(f) is in the image ofπkf.
Letβ ∈πk+nYn represent an element in the kernel. By increasingn, if necessary, we can assume that incl∗(β) is trivial inπk+nC(fn). By Lemma 4.6 (ii) there is a homotopy classα∈π1+k+n(S1∧Xn) such that (S1∧f
n)∗(α) = S1∧β. The homotopy class ˜α = (−1)k+n·(τS1,X
n)∗(α)∈ πk+n+1(Xn∧S
1) then satisfies (fn∧S1)∗( ˜α) =β∧S1, and thus (σn)∗( ˜α)∈πk+n+1Xn+1 hits ι∗(β)∈πk+n+1Yn+1. So the class represented by β in the colimitπkY is in the image ofπkf :πkX −→πkY.
Exactness atπkC(f): If we apply the previous paragraph to the inclusioni:Y −→C(f) instead of f, we see that the sequence
πkY πk(i)
−−−→πkC(f)
πk(incli)
−−−−−→πkC(i) is exact. We claim that the collaps map
∗ ∪p : C(i)∼=CY ∪fCX −→ S1∧X
is a homotopy equivalence, and thus induces an isomorphism of homotopy groups. Since the composite of the homotopy equivalence ∗ ∪p : C(i) −→ S1∧X with the inclusion of C(f) equals the projection
C(f)−→S1∧X, we can replace the groupπ
kC(i) by the isomorphic groupπk(S1∧X) and still obtain an exact sequence.
To prove the claim we define a homotopy inverse
r : S1∧X −→ CY ∪fCX by the formula r([s, x]) = ( [2s, x] ∈CX for 0≤s≤1/2, and [2−2s, f(x)]∈CY for 1/2≤s≤1,
which is to be interpreted levelwise. [specify the homotopiesr(∗ ∪p)'Id and (∗ ∪p)r'Id]
Exactness at πk(S1∧Y): If we apply the previous paragraph to the inclusioni :Y −→C(f) instead off, we see that the sequence
πkY
πk(inclf)
−−−−−−→πkC(f)
πk(incli)
−−−−−→πk(C(incl)) is exact. We claim that the collaps map
C(proj)∼=C(CX∪fY)∪proj(S1∧X) −→ S1∧Y
[define; give details] is a homotopy equivalence, so induces an isomorphism of homotopy groups. Moreover, the composite
S1∧X −−→incl C(proj)−→S1∧Y
is homotopic to the morphismτ∧f :S1∧X−→S1∧Y, whose effect on homotopy groups is the negative of
πk(S1∧f). Since the sign has no effect on the kernel, we can replace the groupπ
kC(proj) by the isomorphic group πk(S1∧Y) and still obtain an exact sequence.
We draw some consequences of Proposition 4.7. For every morphism A −→ B of symmetric spectra which is levelwise an h-cofibration [define] (in the topological context) respectively levelwise injective (in the simplicial context), the quotient spectrum B/A is level equivalent to the mapping cone. Dually, if
f :X −→Y is a morphism of symmetric spectra which is levelwise a Serre fibration of spaces respectively Kan fibration of simplicial sets, the strict fibreF is level equivalent to the homotopy fibre. This gives:
Corollary 4.8. (i) Supposef :A−→B is a h-cofibration of symmetric spectra of topological spaces
or an injective morphism of symmetric spectra of simplicial sets. Denote by p: B −→ B/A the quotient map. Then there is a natural long exact sequence of homotopy groups
· · · −→ πkA πk(f) −−−→ πkB πk(p) −−−→ πk(B/A) δ −−→ πk−1A −→ · · ·
where the connecting mapδis the composite of the inverse of the isomorphismπkC(f)−→πk(B/A)induced
by the level equivalence C(f) −→B/A which collapses the cone of A and the connecting homomorphism πkC(f)−→πk−1A defined in (4.2).
(ii) Supposef : X −→ Y is a morphism of symmetric spectra which is levelwise a Serre fibration of spaces respectively Kan fibration of simplicial sets. Denote by i: F −→X the inclusion of the fibre over the basepoint. Then there is a natural long exact sequence of homotopy groups
· · · −→ πkF πk(i) −−−→πkX πk(f) −−−→ πkY δ −−→πk−1F −→ · · ·
where the connecting map δ is the composite of the connecting homomorphismπkY −→πk−1F(f) defined
in (4.5)and the inverse of the isomorphism πk−1F(f)−→πk−1F induced by the level equivalence F −→
F(f)which sends x∈F to(const∗, x).
(iii)Suppose thatf :X −→Y is an h-cofibration (when in the topological context) respectively levelwise injective and X andY are levelwise Kan complexes (when in the simplicial context). Then the morphism h : S1∧F(f) −→ Y /X (4.4) from the suspension of the homotopy fibre to the quotient of f is a π
∗-
isomorphism.
Proof. (iii) Compare the two long exact sequences and use the five lemma.
Corollary 4.9. (i)For every family of symmetric spectra{Xi}i∈I and every integer k the canonical
map M i∈I πkXi −→ πk _ i∈I Xi !
is an isomorphism of abelian groups.
(ii) For every finiteindexing set I, every family {Xi}
i∈I of symmetric spectra and every integer kthe
canonical map πk Y i∈I Xi ! −→ Y i∈I πkXi
is an isomorphism of abelian groups.
(iii) For every finite family of symmetric spectra the natural morphism from the wedge to the product is aπ∗-isomorphism.
The restriction tofiniteindexing sets in part (ii) of the previous corollary is essential, compare Exam- ple 4.39.
Proof. (i) We first show the special case of two summands. If A and B are two symmetric spectra,
then the wedge inclusion iA:A−→A∨B has a retration. So the associated long exact homotopy group sequence of Proposition 4.7 (i) splits into short exact sequences
0 −→ πkA πk(iA)
−−−−→ πk(A∨B) incl
−−→ πk(C(iA)) −→ 0.
The mapping coneC(iA) is isomorphic to (CA)∨B and thus homotopy equivalent toB. So we can replace
4. HOMOTOPY GROUPS,M-MODULES AND SEMISTABILITY 55