Download SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY

SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
In the previous section, we exploited the interplay between (relative) CW complexes and fibrations to construct the Postnikov and Whitehead towers approximating a given space. In this
context, K(π, n)-spaces naturally came up. In this lecture, we’ll give a few of the most elementary
examples of K(π, n)-spaces, and study the interplay between relative CW complexes and fibrations.
Remark 1. Recall from the previous lecture that a K(π, n)-space, or an Eilenberg-MacLane
space of type (π, n), is a space (X, x0 ) such that πi (X, x0 ) ∼
= ∗ for all i 6= n together with an
isomorphism
πn (X, x0 ) ∼
= π.
Here π can be a pointed set if n = 0, a group is n = 1, or an abelian group if n ≥ 2. It can be shown
that for such a π, a K(π, n)-space always exists, and is unique up to homotopy. We will not give
the general construction in this lecture, but restrict ourselves to the discussion of some examples.
Example 2. (Examples of K(π, n)-spaces)
(1) The circle S 1 is a K(Z, 1)-space. Indeed, it is a connected space with fundamental group
Z, and one way to see that the higher homotopy groups vanish is to consider the universal
covering space R → S 1 . This is a fiber bundle with discrete fiber F and contractible total
space, so the long exact sequence gives us isomorphisms 0 = πi (F ) ∼
= πi+1 (S 1 ) for i > 0.
(2) The same argument applies to wedges of spheres. Consider for example the ‘figure eight’
S 1 ∨ S 1 . Its fundamental group is the free group on two generators Z ∗ Z. The universal
cover of S 1 ∨ S 1 can be explicitly described in terms of the ‘grid’ in the plane,
G = (Z × R) ∪ (R × Z) ⊆ R2 .
The map w : G → S 1 ∨ S 1 can be described by wrapping each edge of length 1 in the grid
around one of the circles (in a way respecting orientations): say the vertical edges to the left
hand circle and the horizontal edges to the right hand one. The universal cover E of S 1 ∨S 1
is the space of homotopy classes of paths in G which start in the origin, and E → S 1 ∨ S 1
is the composition
1
w
E→
G → S1 ∨ S1
(where 1 is evaluation at the endpoint). The fiber of 1 : E → G over a given grid point
(n, m) with n, m ∈ Z is the set of ‘combinatorial paths’ from (0, 0) to (n, m): a sequence of
alternating decisions: go left or go right, go up or go down, where successions of up-down
and left-right cancel each other. Since each homotopy class of paths in E has a unique such
combinatorial description, the space E is clearly contractible.
(3) Recall that RPn , the real projective space of dimension n, is the space of lines in Rn+1 . It
can be constructed as S n / Z2 where the group Z2 = {0, 1} acts by the antipodal map on
the unit sphere
S n = {(x0 , . . . , xn ) ∈ Rn+1 | x20 + . . . + x2n = 1}.
The embedding S n → S n+1 sending (x0 , . . . , xn ) to (x0 , . . . , xn , 0) sends S n to the ‘equator’
inside S n+1 , and is compatible with this antipodal action so that we get a commutative
1
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SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
diagram
S0
/ S1
/ S2
/ ...
RP 0
/ RP 1
/ RP 2
/ ...
There is a ‘natural’ CW decomposition of S n+1 , given inductively by a CW decomposition
of S n with two (n + 1)-cells attached to it: the northern and the southern hemispheres.
This makes S n into a CW complex with exactly two k-cells in each dimension k ≤ n. One
can also take the union along the upper row of the diagram (with the weak topology) to
obtain the infinite-dimensional sphere
[
S∞ =
Sn,
n
a CW complex with exactly two n-cells in each dimension n. Note that since πi (S n ) ∼
= 0 for
∞
i < k we also obtain πi (S ∞ ) ∼
0
for
all
i
≥
0.
In
other
words,
S
is
a
weakly
contractible
=
CW complex, and hence by Whitehead’s theorem is contractible. In a similar way, we can
take the union along the lower row in the above diagram to obtain
[
RP∞ =
RPn ,
n
a CW complex, the infinite-dimensional real projective space, with exactly one n-cell
in each dimension n. The long exact sequence of the covering projection S n → RPn with
discrete fiber Z2 shows that
πi (RPn ) = 0,
1 < i < n, or i = 0,
π1 (RPn ) ∼
= Z2 ,
and by passing to the limit, one concludes that RP∞ is a K(Z2 , 1)-space. (Alternatively,
one can show that S ∞ → RP∞ is still a covering projection with fiber Z2 to conclude that
RP∞ is a K(Z2 , 1).)
(4) Recall that CPn , the complex projective space of (complex) dimension n, is the space of
(complex) lines in Cn+1 . It can be constructed as (Cn+1 −{0})/ C× where C× = C −{0}
acts by multiplication; or, by choosing points on the line of norm 1, as the quotient of the
unit sphere in Cn+1 ,
CPn = S 2n+1 /S 1 ,
where S 1 ⊆ C again acts by multiplication. The quotient S 2n+1 → CPn has enough local
sections (check this!), hence is a fiber bundle with fiber S 1 . The embedding
Cn+1 → Cn+2 : (z0 , . . . , zn ) 7→ (z0 , . . . , zn , 0)
induces maps
S 2n+1
/ S 2n+3
CPn
/ CPn+1
and one can again take the union, to obtain a map S ∞ → CP∞ with CP∞ the infinitedimensional complex projective space. The space CP∞ is a quotient of S ∞ by S 1 , and
the map is again a fiber bundle. The spaces CPn have compatible CW complex structures,
SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
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given by exactly one k-cell in each dimension k ≤ n. One way to see this is to represent a
line in Cn+1 by a point
z = (z0 , . . . , zn ),
zn ∈ R,
and ||z|| = z02 + . . . + zn2 = 1.
zn ≥ 0,
There is a unique way of doing this. Then
p the last coordinate t = zn is uniquely determined
1 − ||z 0 ||), and these (z0 , . . . , zn−1 ) form a disk of
by z 0 = (z0 , . . . , zn−1 ) (since t =
dimension 2n. The boundary of this disk is given by ||z 0 || = 1, in other words t = 0, and
this is exactly the part already in CPn−1 . In any case, either of the two arguments at the
end of the previous example shows that CP∞ is a K(Z, 2)-space.
Next, we will discuss the relation between Serre fibrations and relative CW complexes in terms
of lifting properties. Recall that, by definition, a Serre fibration is a map having the right-liftingproperty (RLP) with respect to inclusions of the form I n × {0} → I n+1 ; or equivalently, with
respect to the inclusion
J n = I n × {0} ∪ ∂I n × I ⊆ I n+1 .
Each of these two kinds of inclusions are relative CW complexes as well as homotopy equivalences.
(In fact, all the spaces involved are contractible.) We will prove a much more general statement in
Theorem 4 below. But first, we state the following lemma, where ‘HELP’ stands for the homotopy
extension and lifting property.
Lemma 3. (HELP) Let p : E → X be a Serre fibration. Then for any relative CW complex
A → B, the map p has the RLP with respect to the inclusion
A × I ∪ B × {0} → B × I.
Proof. Consider a commutative square of solid arrows
α
A × I ∪ B × {0}
q
B×I
q
q
qγ
β
q
/
q8 E
p
/X
S
in which we wish to find a diagonal γ, as indicated. One can write B = n B (n) where A = B (−1)
and B (n) is obtained from B (n−1) by attaching n-cells. It thus suffices to construct larger and larger
liftings γ (n) : B (n) × I → E making the appropriate diagram
γ (n−1) ∪α
/5
k kE
k
k k
p
k k γ (n)
k
k
/X
B (n) × I
β|
B (n−1) × I ∪ B × {0}
commute. Suppose γ (n−1) has been constructed. Then we can find γ (n) by defining it on each of
the n-cells separately, making sure it agrees with γ (n−1) on the boundary of that cell.This reduces
the problem to finding a lift in a diagram of the form
/
o7 E
o
o
p
oo
o
o
/ X.
en × I
∂en × I ∪ en × {0}
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SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
But such a lift exists because the left hand inclusion is essentially (i.e., up to homeomorphism) the
same as J n → I n+1 .
Theorem 4. A map p : E → X is a Serre fibration if and only if it has the RLP with respect to
any inclusion A → B which is at the same time a relative CW complex and a weak equivalence.
Proof. One implication is immediate from the definition of Serre fibrations. Note next that, by
Whitehead’s theorem, such an inclusion i : A → B is a strong deformation retract, i.e., there is a
retraction r : B → A (meaning ri = idA ) and a homotopy h : B × I → B from ir to idB relative
to A. Now suppose we are given a commutative diagram
/E
α
A
p
i
B
/ X.
β
Let γ : B → E be the composition γ = αr. Then γ is not yet a diagonal filler, because one of
the triangles γi = α commutes, but the other doesn’t, in general, since pγ = pαr = βir is only
homotopic to β (via the homotopy βh). We can ‘correct’ this by invoking HELP, and finding a
lift k in the commutative (!) square
απ1 ∪γ
5/
k kE
k
k k
p
k k k
k
k
/ X.
B×I
βh
A × I ∪ B × {0}
Then γ 0 = k(−, 1) is the required map filling the first diagram, since pγ 0 = β while γ 0 i = α as one
easily checks.
Theorem 4 states that in a square of the form
/E
α
A
p
i
B
/ X.
β
in which p is a Serre fibration while i is a relative CW complex, one can find a diagonal in case i
is a weak homotopy equivalence. In fact, the situation is symmetric in the sense that one can also
ask p to be a weak homotopy equivalence.
Theorem 5. Let p : E → X be a Serre fibration and a weak homotopy equivalence. Then p has the
RLP with respect to any relative CW complex A → B.
For the proof, we use the following easy lemma.
Lemma 6. Let
/E
Y ×X E
p
π1
Y
f
/X
SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
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be a pullback square. If p : E → X is a Serre fibration and a weak homotopy equivalence then so is
its pullback π1 along f .
Proof. We already know from an earlier lecture that the pullback of a Serre fibration is again one.
The fact that π1 is a weak homotopy equivalence whenever p is now easily follows from the long
exact sequence of a fibration. Indeed, p is a weak homotopy equivalence if and only its fiber F
is weakly contractible and if p is surjective on path components. Using that the fiber of π1 is
homeomorphic to that of p one easily concludes.
Proof. (of Theorem 5) Suppose p : E → X is a Serre fibration and a weak homotopy equivalence.
Given a commutative square
α /
E
A
p
B
/ X.
β
where A → B is a relative CW complex, one can try to construct a diagonal filler by ‘induction over
the cells’, exactly as in the proof of HELP, and this reduces the problem to the case where A → B
is the inclusion i : ∂en → en of the boundary of a cell. So, suppose we are given commutative square
of the form
α /
E
∂en
z=
z
p
i
z
z
/ X.
en
β
Finding a diagonal lift in this diagram is the same as finding one in the square on the left
∂en
i
(i,α)
tt
en
t
/ en ×X E
t:
t π
1
π2
p
/ en
id
/E
β
/X
where the right hand square is the pullback (check this!). Write p0 : E 0 → en for π1 : en ×X E → en ,
so that the square on the left looks like
∂en
i
z
en
α0
z
z
=
/ E0
z=
p0
/ en
with α0 = (i, α). Then α0 represents an element of πn−1 (E 0 ) which must be zero because p0 is a weak
homotopy equivalence (by the lemma) and en is contractible. So α0 extends to a map γ : en → E 0
– ‘extends’ in the sense that γi = α0 –, but we do not know whether p0 γ = iden . On the other
hand, p0 γ is homotopic to the identity by a homotopy relative to the boundary ∂en , which can be
described explicitly as the ‘convex combination’
h(x, t) = tx + (1 − t)pγ 0 (x).
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SECTION 12: HOMOTOPY EXTENSION AND LIFTING PROPERTY
We now invoke HELP again, exactly as in the proof of Theorem 4: choose a lift in
α0 π1 ∪γ
5/ 0
k kE
k
k k
p0
k k k
k
k
/ en
en × I
h
∂en × I ∪ en × {0}
and let γ 0 = k(−, 1). Then γ 0 i = α and p0 γ 0 = iden , so γ 0 is the required lift in the previous
diagram.
Remark 7. Theorem 4 and Theorem 5 represent the most important part of Quillen’s famous
axioms for a (closed) model category, about which we will say more in one of the remaining
lectures.