Simultaneous Z/p-acyclic resolutions of expanding sequences

0.29 MB·English

by  Leonard R. Rubin

Z/p-ACYCLIC RESOLUTIONS IN THE STRONGLY COUNTABLE
Z/p-DIMENSIONAL CASE
LEONARDR.RUBIN,VERATONIC´
2
1
0
2 Abstract. Weprovethe following theorem.

g Theorem: LetX beanonempty compactmetrizablespace, letl1 ≤l2 ≤...beasequence
u in N, and let X1 ⊂ X2 ⊂ ... be a sequence of nonempty closed subspaces of X such that
A for each k ∈N, dimZ/pXk ≤ lk. Then there exists a compact metrizable space Z, having
closed subspaces Z ⊂Z ⊂..., and a (surjective) cell-like map π:Z →X, such that for
1 2
3 each k∈N,
2
(a) dimZ ≤l ,
k k
(b) π(Z )=X , and
] k k
T (c) π|Zk :Zk →Xk is a Z/p-acyclic map.
G Moreover, there is a sequence A1 ⊂ A2 ⊂ ... of closed subspaces of Z such that for each
k, dimA ≤l , π| :A →X is surjective, and for k∈N, Z ⊂A and π| :A →X
. k k Ak k k k Ak k
h is a UVlk−1-map.
at It is not required that X = S∞k=1Xk or that Z = S∞k=1Zk. This result generalizes
m the Z/p-resolution theorem of A. Dranishnikov and runs parallel to a similar theorem of
S.Ageev,R.Jim´enez,andthefirstauthor,whostudiedthesituationwherethegroupwas
[
Z.

2
v
1. Introduction
0
8
The goal of this paper is to prove the following theorem.
4
2
Theorem 1.1. Let X be a nonempty compact metrizable space, let l ≤ l ≤ ... be a
. 1 2
1
sequence in N, and let X ⊂ X ⊂ ... be a sequence of nonempty closed subspaces of X
0 1 2
such that for each k ∈N, dim X ≤ l . Then there exists a compact metrizable space Z,
1 Z/p k k
1 having closed subspaces Z ⊂ Z ⊂ ..., and a (surjective) cell-like map π : Z → X, such
1 2
v: that for each k ∈N,
Xi (a) dimZk ≤ lk,
(b) π(Z )= X , and
r k k
a (c) π| : Z → X is a Z/p-acyclic map.
Zk k k
Moreover, there is a sequence A ⊂ A ⊂ ... of closed subspaces of Z such that for each k,
1 2
dimA ≤ l , π| : A → X is surjective, and for k ∈ N, Z ⊂ A and π| : A → X is a
k k Ak k k k Ak k
UVlk−1-map.
The second Section will contain some technical results necessary for the proof of Theo-
rem 1.1, and the proof will be described in the third Section.
In Section 4 we will outline a proof of a case of Theorem 1.1 that was suggested to us
by an anonymous referee. Unfortunately, this technique cannot be used to prove the most
difficult cases of Theorem 1.1, nor does it have the potential for generalization for those
groups G whose resolutions require a domain space of dimension n+1, if the range space
had dim ≤ n ([Le]).
G
Date: 22 August,2012.
2010 Mathematics Subject Classification. Primary 55M10, 54F45; Secondary 55P20.
Keywordsandphrases. cell-likemap,cohomologicaldimension,CW-complex,dimension,Edwards-Walsh
resolution, Eilenberg-MacLane complex, G-acyclic map, inverse sequence, simplicial complex, UVk-map.
1
2 L.Rubin,V.Toni´c
For example, the theorem that follows is an immediate consequence of Theorem 1.1, but
it cannot be proven using the technique described in Section 4.
Theorem 1.2. Let n ∈ N and let (X ) be a sequence of (not necessarily nested) closed
i
subsets of the Hilbert cube Q with dim X ≤ n for all i. Then there exists a compact
Z/p i
metrizable space Z, a cell-like map π : Z → Q, and a sequence (Z ) of closed subsets of Z
i
such that ∀i,
(a) dimZ ≤n, and
i
(b) π| : Z → X is a surjective Z/p-acyclic map.
Zi i i
This theorem provides a cell-like resolution of the Hilbert cube Q and simultaneously
Z/p-acyclic resolutions over any F -collection whatsoever of such X .
σ i
Let us proceed by explaining some terms that might be unfamiliar to the reader. Basic
facts about cell-like spaces and maps can be found in [Da]. A map π : Z → X is called
cell-like if for each x ∈ X, π−1(x) has the shape of a point. To detect that a compact
metrizable space Y has the shape of a point, it is sufficient to prove that there is an inverse
sequence (Z ,pi+1), of compact metrizable spaces Z , whose limit is homeomorphic to Y
i i i
and such that for infinitely many i ∈ N, pi+1 : Z → Z is null-homotopic. It is also
i i+1 i
sufficient to show that every map of Y to a CW-complex is null-homotopic.
A map π : Z → X between topological spaces is called G-acyclic ([Dr]) if all its fibers
π−1(x) have trivial reduced Cˇech cohomology with respect to a given abelian group G, or,
equivalently, every map f : π−1(x) → K(G,n) is null-homotopic.
Note that a map π : Z → X being cell-like implies that π is also G-acyclic.
To detect that a compact metrizable space Y has trivial reduced Cˇech cohomology with
respect to the group G, it is sufficient to prove that there is an inverse sequence (Z ,pi+1)
i i
of compact polyhedra Z whose limit is homeomorphic to Y, such that for infinitely many
i
i ∈ N, the map pi+1 : Z → Z induces the zero-homomorphism of cohomology groups
i i+1 i
Hm(Z ;G) → Hm(Z ;G), for all m ∈ N.
i i+1
A map π : Z → X is called a UVk-map ([Da]) if each of its fibers has property UVk.
This means that each embedding π−1(x) ֒→ A into an ANR A has property UVk: for every
0 ≤ r ≤ k and every neighborhood U of π−1(x) in A, there exists a neighborhood V of
π−1(x) in U such that every map of Sr into V is null-homotopic in U. In order to prove
that π is a UVk-map, it is sufficient to show that, for all x ∈ X, there is an inverse sequence
(Z ,pi+1) of compact polyhedra Z , whose limit is homeomorphic to π−1(x) and such that
i i i
∀i ∈ N, if 0 ≤ r ≤ k, then any map h : Sr → Z is null-homotopic. It is well-known that
i
cell-like compacta have property UVk for all k.
A map g :X → |K| between a space X and a polyhedron |K| is called a K-modification
of a map f : X → |K| if whenever x ∈ X and f(x) ∈ σ, for some σ ∈ K, then g(x) ∈ σ.
◦
This is equivalent to the following: whenever x ∈ X and f(x) ∈ σ, for some σ ∈ K, then
g(x) ∈ σ.
The proof of Theorem 1.1 uses some techniques developed by A. Dranishnikov in the
proof of the following theorem, which can be found as Theorem 8.7 in [Dr].
Theorem 1.3. For every compact metrizable space X with dim X ≤ n, there exists a
Z/p
compact metrizable space Z and a surjective map π : Z → X such that π is Z/p-acyclic and
dimZ ≤ n.
We will show in Remark 3.3 that our Theorem 1.1 is a generalization of this theorem.
Dranishnikov used Edwards–Walsh complexes and resolutions, and so shall we.
The following definition of Edwards–Walsh complexes (EW-complexes) and resolutions,
as well as results about them, can be found in [Dr], [DW] or [KY]. For G = Z, these
resolutions were formally formulated in [Wa].
Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case 3
Definition1.4. LetGbeanabeliangroup, n ∈ NandLasimplicialcomplex. An Edwards–
Walshresolution of L indimension nisapair (EW(L,G,n),ω) consisting of a CW-complex
EW(L,G,n) and a combinatorial map ω : EW(L,G,n) → |L| (that is, ω−1(|L′|) is a
subcomplex, for each subcomplex L′ of L) such that:
(i) ω−1(|L(n)|) = |L(n)| and ω| is the identity map of |L(n)| onto itself,
|L(n)|
(ii) for every simplex σ of L with dimσ > n, the preimage ω−1(|σ|) is an Eilenberg–
MacLane space of type ( G,n), where the sum G is finite, and
(iii) for every subcomplex L′Lof L and every map f :L|L′| → K(G,n), the composition
f ◦ω| :ω−1(|L′|) → K(G,n) extends to a map F :EW(L,G,n) → K(G,n).
ω−1(|L′|)
We usually refer to the CW-complex EW(L,G,n) as an Edwards–Walsh complex, and to
the map ω itself as an Edwards–Walsh projection.
Remark 1.5. Let L′ be a subcomplex of L, K be the subcomplex ω−1(|L′|) of EW(L,G,n),
and ω = ω| : ω−1(|L′|) → |L′|. Then (K,ω ) is an Edwards–Walsh resolution of
L′ ω−1(|L′|) L′
the form (EW(L′,G,n),ω ).
L′
Adiscussion abouttheexistence of Edwards–Walsh resolutions, as well as their construc-
tion, can befound in [Dr], [DW], [KY], [Wa]. For our needs, it is enough to know that when
G is either Z or Z/p, Edwards–Walsh resolutions exist for any simplicial complex L.
In particular, we shall briefly describe the construction of (EW(L,Z/p,n),ω) for a finite-
dimensional simplicial complex L. If dimL ≤ n, define EW(L,Z/p,n) = L(n) = L, and
ω = id . If dimL = n+1, we start with L(n) and the identity map id , and proceed by
L L(n)
building a K(Z/p,n) on ∂σ, for each (n+1)-simplex σ of L, and we build ω by extending
∂σ ֒→ σ over this newly attached K(Z/p,n). In this way, ω−1(|σ|) = K(Z/p,n), ∀ (n+1)-
simplex σ of L.
If dimL > n+1, then we shall distinguish the cases n ≥ 2 and n = 1. In both of these
cases our construction is inductive.
If n ≥2 anddimL > n+1, then theskeleton L(n+1) is dealt with as describedabove, i.e.,
by attaching a K(Z/p,n) to ∂σ, for each (n+1)-simplex σ ∈ L(n+1). This represents the
basisof ourinductiveconstruction. For thestepof ourinductiveconstruction, let k > n+1.
Then for any k-simplex σ of L, we have that π (ω−1(|∂σ|)) = Z/p, where this sum is
n
finite. So ω−1(|σ|) will be obtained from ω−1(|∂σ|) by attachinLg cells of dim ≥ n+ 2 in
order to kill off the higher homotopy groups of ω−1(|∂σ|), and achieve that ω−1(|σ|) =
K( Z/p,n).
ILf n = 1 and dimL > 2, then the 2-skeleton L(2) is dealt with as described above, that
is, by attaching a K(Z/p,1) to ∂σ, for each 2-simplex σ ∈ L(n+1). To be more precise, this
means attaching a 2-cell using a map of degree p from the boundary of the 2-cell to ∂σ,
for every 2-simplex σ of L, and then proceeding by attaching cells of dim ≥ 3 to form a
K(Z/p,1) on top of each of these Moore spaces. However, the above mentioned 2-cells are
not the only ones that get attached here, we will have to attach more of these. Namely,
when k > 2, then for any k-simplex σ of L, there will be 2-cells γ ⊂ ω−1(|σ|)\ω−1(|∂σ|),
attached by a map ∂γ → ω−1(|∂σ|) representing a commutator in π (ω−1(∂σ)). This is
1
to ensure that π (ω−1(|σ|)) = Z/p. We proceed by attaching cells of dimension ≥ 3 to
1
achieve that ω−1(|σ|) = K( ZL/p,1).
L
The following fact is proven in [Dr] as Lemma 8.1, and (iv ) is clear from our construc-
Z/p
tion above.
Lemma 1.6. For the groups Z and Z/p, for any n ∈ N and for any simplicial complex L,
there is an Edwards–Walsh resolution ω : EW(L,G,n) → |L| with the additional property
for n > 1:
4 L.Rubin,V.Toni´c
(iv ) the (n+1)-skeleton of EW(L,Z,n) is equal to L(n);
Z
(iv ) the (n+1)-skeleton of EW(L,Z/p,n) is obtained from L(n) by attaching (n+1)-cells
Z/p
by a map of degree p to the boundary ∂σ, for every (n+1)-dimensional simplex σ.
Here are some other properties following from the construction of Edwards-Walsh com-
plexes for the groups Z/p.
Remark 1.7. Let L be a simplicial complex, let σ be any simplex of L with dimσ > n,
and let (EW(L,Z/p,n),ω) be an Edwards-Walsh resolution of L. According to Remark 1.5,
ω−1(|σ|) = EW(σ,Z/p,n) and from the construction of EW(L,Z/p,n), we have that the
number of summands in π (ω−1(|σ|)) ∼= Z/p is less than or equal to the number of the
n
(n+1)-faces of σ. L
From this Remark and our construction, we get:
Corollary 1.8. Let σ be a simplex with dimσ > n, taken as a simplicial complex, and let
(EW(σ,Z/p,n),ω) be an Edwards-Walsh resolution of σ. Then
(I) H (|σ(n)|)∼= rZ, and
n 1
(II) Hn(EW(σ,Z/Lp,n)) ∼= r1Z/p,
where r ≤ the number of all (Ln+1)-faces of σ. Moreover,
(III) we can choose τ ,...,τ to be some (n+1)-faces of σ so that the images h ,...,h
1 r 1 r
of the generators of H (∂τ ),...,H (∂τ ), induced by the inclusions ∂τ ֒→ σ(n),
n 1 n r i
form a basis of H (|σ(n)|). Then if q ,...,q are the images of the generators of
n 1 r
H (∂τ ),...,H (∂τ ), induced by the inclusions ∂τ ֒→ EW(σ,Z/p,n), and λ =
n 1 n r i ∗
H (λ) is induced by the inclusion λ : σ(n) ֒→ EW(σ,Z/p,n), we get that q =
n 1
λ (h ),...,q = λ (h ) form a basis of H (EW(σ,Z/p,n)).
∗ 1 r ∗ r n
The following lemma is proven in [Dr] as Lemma 8.2. It concerns (approximately) lifting
maps through EW-complexes:
Lemma 1.9. Let X be a compact metrizable space with dim X ≤ n, and let L be a finite
G
simplicial complex. Then for every Edwards–Walsh resolution ω : EW(L,G,n) → |L|, and
for every map f :X → |L|, there exists a map f′ :X → EW(L,G,n) such that
(i) f′| = f| , and
f−1(|L(n)|) f−1(|L(n)|)
(ii) ω◦f′ is an L-modification of f.
Our primary construction will be done in the Hilbert cube Q – our space X is compact
metrizable, and Q is universal for all compact metrizable spaces.
∞
Let the Hilbert cube Q = I be endowed with the metric ρ such that if x = (x ),
i
Yi=1
∞
|x −y |
i i
y = (y ), then ρ(x,y) = . As usual, I = [0,1]. For any i ∈ N it will be
i 2i
Xi=1
convenient to write Q = Ii × Q in factored form. In this case, any subset E of Ii will
i
always be treated as E ×{0} ⊂ Q. We shall use p : Q → Ii for coordinate projection.
i
In some of the proofs that follow we will use stability theory, about which more details
canbefoundin§VI.1of[HW]. Namely, wewillusetheconsequencesofTheoremVI.1. from
[HW]: ifX isa separablemetrizable spacewith dimX ≤ n, thenfor any mapf : X → In+1
all values of f are unstable. A point y ∈ f(X) is called an unstable value of f if for every
δ > 0 there exists a map g :X → In+1 such that:
(1) d(f(x),g(x)) < δ for every x ∈ X, and
(2) g(X) ⊂ In+1\{y}.
Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case 5
Moreover, this map g can be chosen so that g = f on the complement of f−1(U), for any
open neighborhood U of y, and so that g is homotopic to f (see Corollary I.3.2.1 of [MS]).
The following lemma is a form of the homotopy extension theorem with control, and can
be found in [AJR] as Lemma 2.1.
Lemma 1.10. Let f : X → R be a map of a compact polyhedron X to a space R, X be
0
a closed subpolyhedron of X, and U be an open cover of R. Suppose that F : X ×I → R
0
is a U-homotopy of f| . Then there exists a U-homotopy H : X ×I → R of f such that
X0
H| =F :X ×I → R.
X0×I 0
Notation. We will use the following notation. Let x belong to a metric space X and let
δ > 0. Then by N(x,δ) we shall mean the closed δ-neighborhood of x in X. For example,
for x ∈ Q, p (x) ∈ Ii so N(p (x),δ) is the closed δ-neighborhood of p (x) in Ii.
i i i
Whenever (P ,gi+1) is an inverse sequence, T ⊂ P and gi+1(T ) ⊂ T for each i, then
i i i i i i+1 i
we shall write (T ,gi+1) for the induced inverse sequence, using the same notation for the
i i
bonding maps as long as no confusion can arise.
Whenever P is a polyhedron, τ is a triangulation of P, and k ≥ 0, then P(k) will denote
the subpolyhedron of P triangulated by the k-skeleton of τ, i.e., P(k) = |τ(k)|. If R is a
subpolyhedron of P and we have to build an Edwards-Walsh complex on τ| , we will write
R
EW(R,Z/p,n) instead of EW(τ| ,Z/p,n), to keep matters simpler.
R
2. Technical lemmas
The following type of result is a lemma which is technical, but which will help us find
certain mapsandunderstandtheirfibers. Thislemmacan befoundin[AJR]as Lemma3.1.
Once the correct conditions are found on the construction of said maps, then Theorem 1.1
will follow readily.
Lemma 2.1. Suppose that for each i ∈ N we have selected n ∈ N, a compact subset
i
Pi ⊂ Ini, δi > 0, εi > 0, and a map gii+1 : Pi+1 → Pi so that:
(i) if u, v ∈ Q and ρ(u,v) ≤ ε , then ρ(p (u),p (v)) < δ ,
i+1 ni ni i
(ii) n < n ,
i i+1
(iii) 9 < ε ,
2ni i
(iv) ρ(gi+1(x),p (x)) < δ for all x ∈ P ,
i ni i i+1
(v) δ < 1 , and
i 2ni−1
(vi) P ×Q ⊂ P ×Q .
i+1 ni+1 i ni
∞
Put X = P ×Q , P = (P ,gi+1), and Z = limP. Then for each z = (a ,a ,...) ∈
i ni i i 1 2
i\=1
∞
Z ⊂ P , and associated sequence (a ) in Q,
i i
Yi=1
(a) (a ) is a Cauchy sequence in Q whose limit lies in X, and
i
(b) the function π : Z → X given by π(z) = lim(a ) is continuous.
i
i→∞
Fix x ∈ X and for each i ∈ N, let B = N(p (x),2δ )∩P ,B# = N(p (x),ε )∩P .
x,i ni i i x,i ni i i
Then,
(c) B ⊂ B# and gi+1(B# )⊂ B .
x,i x,i i x,i+1 x,i
If we let P = (B ,gi+1) and P# = (B#,gi+1), then,
x x,i i x x,i i
(d) limP = limP#, and
x x
(e) π−1(x) = limP .
x
6 L.Rubin,V.Toni´c
In addition, suppose we are given, for each i ∈ N, a closed subspace T ⊂ P in such a
i i
manner that gi+1(T ) ⊂ T . Put T = (T ,gi+1) and Z′ = limT ⊂ Z. For x ∈ X, let
i i+1 i i i
S = B ∩T , T = (S ,gi+1); set π˜ = π| : Z′ → X. Then,
x,i x,i i x x,i i Z′
(f) π˜−1(x) = limT , and
x
(g) if S 6= ∅ for each i, then π˜−1(x) 6= ∅.
x,i
A helpful diagram for Lemma 2.1:
gi+1
i
... oo P(cid:127)_i ssoo pni| Pi(cid:127)+_ 1 oo ... Z✤
✤
π
✤
... oo ?_P ×(cid:15)(cid:15) Q oo ?_P ×Q(cid:15)(cid:15) oo ?_ ... X(cid:15)(cid:15)
i ni i ni+1
Beforeproceeding,notethatifLisasimplicialcomplex, K aCW-complex, andf :|L| →
K a map, then we say that f is cellular if it is cellular with respect to the CW-structure
induced on |L| by L and the given one of K, i.e., f takes the (simplicial) n-skeleton of L to
the (CW) n-skeleton of K, ∀n.
The following Corollary is a version of Corollary 3.2 from [AJR], adapted for the Z/p-
case. When used (in the proof of the main theorem), A can be replaced by Z (not just
k k
by A of Theorem 1.1).
k
Corollary 2.2. Suppose in Lemma 2.1 that for each i ∈ N, P = |τ | is a nonempty
i i
subpolyhedron of Ini having a triangulation τi, with a subdivision τi with meshτi < δi, so
that for every simplex γ of τ , τ | is collapsible. Moreover, assume that gi+1 is a simplicial
i i γ i
map (in particular, for all k ≥ 0, gi+1(P(k)e) ⊂ P(k), where τ and τ are the relevant
e i i+1 i i+1 i
triangulations). Let l ≤ l ≤ ... be a sequence in N, and let
1 2
T =(P(lk),gi+1), and A = limT .
k i i k k
Then A ⊂ A ⊂ ..., and for each k ≥ 1,
1 2
(I) dimA ≤ l and π| :A → X is surjective.
k k Ak k
Assume further that for each x ∈ X and i∈ N, there is a contractible polyhedron P which
x,i
is the closed star of a vertex in the triangulation τ , such that
i
#
B ⊂ P ⊂ B .
x,i x,i e x,i
Then
(II) π :Z → X is a cell-like map, and
(III) for each k ∈ N, π| : A → X is a UVlk−1-map.
Ak k
Suppose now that all of the above statements are true, and let k ∈N. If for infinitely many
indexesiwehavethatforallx ∈ X, ω◦f¯(P )⊂ P , andgi+1| ≃ ω◦f¯| , where
i x,i+1 x,i i Px,i+1 i Px,i+1
ω :EW(P ,Z/p,l ) → P is an Edwards–Walsh projection, and f¯ :P → EW(P ,Z/p,l )
i k i i i+1 i k
is a cellular map, then
(IV) π| : A → X is a Z/p -acyclic map.
Ak k
Before showing the proof of Corollary 2.2, we will state and prove some lemmas which
will be useful for its proof.
Lemma 2.3. Let n ∈ N, and let P = |L| and Q = |M| be compact polyhedra with dimP,
dimQ ≥ n+1. For any (n+1)-simplex τ of M, let h and q be the images of a generator
e e e
of H (∂τ ) under the maps of H (∂τ ) induced by the inclusions ∂τ ֒→ |M(n)| and ∂τ ֒→
n e n e e e
EW(M,Z/p,n), respectively.
Let µ, ν and λ be the inclusions as shown in the upcoming diagram, and let f : |L| →
EW(M,Z/p,n) be a cellular map making this diagram commutative.
Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case 7
Moreover, let M be such that:
(I) H (|M(n)|) ∼= rZ, and
n 1
(II) Hn(EW(M,Z/Lp,n)) ∼= r1Z/p,
where r ≤ the number of all (nL+1)-simplexes of M; and
(III) we can choose some (n+1)-simplexes τ ,...,τ of M so that {h ,...,h } forms a
1 r 1 r
basis of H (|M(n)|), and so that {q ,...,q } forms a basis of H (EW(M,Z/p,n)).
n 1 r n
Then for any (n+1)-simplex σ ∈ L, with H (∂σ) = hgi, we have:
n
(a) f ◦ν ◦µ is null-homotopic, so
(b) H (f| ◦µ)(g) = r ε h , where ε ≡ 0 (mod p), for e ∈ {1,...,r} .
n |L(n)| e=1 e e e
P
EW(M,Z/p,n)
|L| ✐✐✐✐✐✐✐✐✐✐✐f✐✐✐✐✐✐✐✐✐44 |M(cid:15)(cid:15) ω|__❃❃❃❃❃❃❃❃❃❃λ
❃
bb❉❉❉❉❉❉ν❉❉0 P gg❖❖❖❖❖❖❖❖❖❃❃❖❃❖❃❖❃4❃T❃/ O
|L(n)| // |M(n)|
OO f|
µ
(cid:31)?
∂σ
Proof: Since ∂σ is contained in σ, which is contractible, the inclusion ν ◦µ : ∂σ ֒→ |L| is
null-homotopic. Therefore f ◦ν ◦µ is null-homotopic, so (a) is true.
To prove (b), notice that f being a cellular map implies
f(|L(n)|) ⊂ EW(M,Z/p,n)(n) = |M(n)|.
It is clear that f ◦ν ◦µ = λ◦f| ◦µ. So (a) implies
|L(n)|
0 = H (f ◦ν ◦µ)(g) = H (λ◦f| ◦µ)(g).
n n |L(n)|
From (III) we get that H (f| ◦µ)(g) = r ε h , for some ε ∈Z, and therefore
n |L(n)| e=1 e e e
P r r
H (λ◦f| ◦µ)(g) = H (λ)( ε h ) = ε q = 0,
n |L(n)| n e e e e
Xe=1 Xe=1
which means that ε ≡ 0 (mod p), ∀e∈ {1,...,r}. (cid:3)
e
Some form of the following lemma was used by various authors.
Lemma 2.4. Let n ∈ N, P = |L| be a compact polyhedron with dimP ≥ n+1 and M be
the closed star of a vertex from L(0). Let L be a subdivision of L such that for every simplex
e f
σ of L, L| is a collapsible simplicial complex. Let M be the simplicial complex that L
|σ| e e
inducees on |M|, i.e., M = L||Mf| (subdivided vertex star). Then
(I) H (|Mf(n)|) ∼= rZ, and
n 1
(II) Hn(EW(M,Z/Lp,n)) ∼= r1Z/p,
where r ≤ the number of all (nL+1)-simplexes of M. Moreover,
(III) we can choose τ ,...,τ to be some (n + 1)-simplexes of M so that the images
1 r
h ,...,h ofthegeneratorsofH (∂τ ),...,H (∂τ ), inducedbytheinclusions∂τ ֒→
1 r n 1 n r i
M(n), form a basis of H (|M(n)|). Then if q ,...,q are the images of the gener-
n 1 r
ators of H (∂τ ),...,H (∂τ ), induced by the inclusions ∂τ ֒→ EW(M,Z/p,n),
n 1 n r i
and H (λ) is induced by the inclusion λ : M(n) ֒→ EW(M,Z/p,n), we get that
n
q = H (λ)(h ),...,q = H (λ)(h ) form a basis of H (EW(M,Z/p,n)).
1 n 1 r n r n
8 L.Rubin,V.Toni´c
Wewillomittheprooftosavespace. OnthewaytoprovingthisLemma,onecanfirstuse
Corollary 1.8 (containing thestatement analogous to this one, butfor a simplex)in order to
prove analogous statements for a (non-subdivided) vertex star, and then for a subdivided
simplex with a collapsible subdivision.
Then Lemma 2.4 can be proven by first proving its statement for dimM = n+1, and
then, by induction, showing it is true for dimM = n+k+1. The general step of induction
would utilize another induction, on the number of (n+k+1)-simplexes of M, as well as a
Mayer-Vietoris sequence. We used a collapsible subdivision on simplexes of M so that we
f
couldorganizetheprocessofinduction. Theinformationabouttheexistenceofsubdivisions
f
of a triangulation on a simplicial complex, in which a simplex with a new subdivision is still
collapsible can be found in [Gl].
Remark 2.5. When M is a subdivided vertex star from Lemma 2.4, then Lemma 2.3 is
true for Q = |M| and |M(n)| is (n−1)-connected.
Proof of Corollary 2.2: Surely dimA ≤ l . Let x ∈ X. Apply Lemma 2.1 with T = P(lk)
k k i i
and
S = B ∩P(lk).
x,i x,i i
Then T becomes T and
k
Z′ = limT = lim(P(lk),gi+1) =A .
k i i k
Note that the representation of X implies that p (X) ⊂ P , ∀i ∈ N. This fact, together
ni i
with meshτ < δ , can be used to check that B must contain a vertex of τ , so S 6= ∅.
i i x,i i x,i
Therefore (g) of Lemma 2.1 shows that (I) is true.
Part (c) of Lemma 2.1 and the fact that B ⊂ P ⊂ B# ∀i ∈ N, show that ∀i ∈ N,
x,i x,i x,i
gi+1(P ) ⊂ P , so P′ := (P ,gi+1) is an inverse sequence. Clearly (see (d) and (e) of
i x,i+1 x,i x x,i i
Lemma 2.1), limP′ = π−1(x). Now P′ is an inverse sequence of contractible polyhedra.
x x
Hence (II) is true.
To get at (III), first observe that by (f) of Lemma 2.1, the fiber (π| )−1(x) is the limit
Ak
of the inverse sequence (S ,gi+1). On the other hand, for each i ∈ N, B ⊂ P ⊂ B#,
x,i i x,i x,i x,i
gi+1(P(lk)) ⊂ P(lk), and gi+1(B# ) ⊂ B . So one deduces that
i i+1 i i x,i+1 x,i
gi+1(P(lk) ) ⊂ gi+1(B# )∩gi+1(P(lk)) ⊂ B ∩P(lk) ⊂ P ∩P(lk) = P(lk).
i x,i+1 i x,i+1 i i+1 x,i i x,i i x,i
Thus P′(lk) := (P(lk),gi+1) is an inverse sequence of compact polyhedra. Since S ⊂ P(lk)
x x,i i x,i x,i
and
gi+1(P(lk) ) ⊂ B ∩P(lk) = S ,
i x,i+1 x,i i x,i
it is clear that limP′(lk) is the same as the limit of the inverse sequence (S ,gi+1), i.e.,
x x,i i
that
(π| )−1(x) = lim (S ,gi+1) = lim (P(lk),gi+1).
Ak x,i i x,i i
We shall show that for each i ∈ N, if 0 ≤ r ≤ l −1 and h : Sr → P(lk) is a map, then h
k x,i
is homotopic to a constant map. Since dimSr = r < l , h is homotopic in P(lk) to a map
k x,i
that carries Sr into P(lk−1) (see remark about stability theory). But P is contractible,
x,i x,i
so the inclusion P(lk−1) ֒→ P(lk) is null-homotopic. This shows that h : Sr → P(lk) is
x,i x,i x,i
null-homotopic. So all fibers of π| are UVlk−1.
Ak
To prove (IV), we need to show that any fiber of π| is Z/p -acyclic, i.e., for infinitely
Ak
many indexes i, the map gi+1| : P(lk) → P(lk) induces the zero-homomorphism of
i P(lk) x,i+1 x,i
x,i+1
Z/p-acyclic resolutions in the strongly countable Z/p-dimensional case 9
cohomology groups Hm(P(lk);Z/p) → Hm(P(lk) ;Z/p), for all m ∈ N (we need not worry
x,i x,i+1
about m = 0 because the P(lk)’s are (l −1)-connected, so their reduced zero-cohomology
x,i k
groupsaretrivial). Wewillbefocusingonthoseindexesiforwhichgi+1| ≃ ω◦f¯| ,
i Px,i+1 i Px,i+1
as mentioned in the conditions of Corollary 2.2.
It is, in fact, enough to show that the map gi+1| : P(lk) → P(lk) induces the zero-
i P(lk) x,i+1 x,i
x,i+1
homomorphism of homology groups with Z/p-coefficients. Here is why this is true. Notice
that each of P and P is a closed vertex star (in the coarser triangulation), subdivided
x,i+1 x,i
so that each original simplex of the vertex star is collapsible as a simplicial complex. So
Lemma2.4(for n= l )istrueforboth|M|= P and|M| = P . Thereforeproperty(I)
k x,i+1 x,i
of Lemma 2.4 is true for both P(lk) and P(lk), and both are (l −1)-connected. Therefore
x,i+1 x,i k
by the Universal Coefficients Theorem for homology and cohomology we have
H (P(lk) ;Z/p)∼= H (P(lk) )⊗Z/p, ∀m ≥ 1, and
m x,i+1 m x,i+1
Hm(P(lk) ;Z/p) ∼= Hom(H (P(lk) ),Z/p), ∀m ≥ 1,
x,i+1 m x,i+1
and for P(lk) analogously, and these expressions are non-zero only for m = l . So if
x,i k
gi+1| : P(lk) → P(lk)inducesthezero-homomorphismH (gi+1;Z/p) :H (P(lk) ;Z/p)→
i P(lk) x,i+1 x,i lk i lk x,i+1
x,i+1
H (P(lk);Z/p), then for any ϕ ∈ Hom(H (P(lk)),Z/p), we have ϕ ◦ H (gi+1) = 0 ∈
lk x,i lk x,i lk i
Hom(Hlk(Px(l,ik+)1),Z/p),thatis,theinducedhomomorphismHlk(gii+1;Z/p) :Hlk(Px(l,ik);Z/p) →
Hlk(P(lk) ;Z/p) is the zero-homomorphism.
x,i+1
So let us show that H (gi+1;Z/p) : H (P(lk) ;Z/p) → H (P(lk);Z/p) is the zero-
lk i lk x,i+1 lk x,i
homomorphism. Before proceeding, note that by Remark 1.5, given an EW-resolution
ω :EW(P ,Z/p,l ) → P , we know that ω−1(P )= EW(P ,Z/p,l ), so
i k i x,i x,i k
ω| :EW(P ,Z/p,l ) → P is also an EW-resolution.
ω−1(Px,i) x,i k x,i
Let σ be any (l + 1)-simplex of P , and let g be a generator of H (∂σ). Let
k x,i+1 σ lk
µ : ∂σ ֒→ P(lk) , ν : P(lk) ֒→ P , and λ : P(lk) ֒→ EW(P ,Z/p,l ) be inclusions.
x,i+1 x,i+1 x,i+1 x,i x,i k
Notice that ω◦f¯(P ) ⊂ P implies that f¯(P ) ⊂ EW(P ,Z/p,l ), and since f¯ is
i x,i+1 x,i i x,i+1 x,i k i
a cellular map, we also have f¯i(Px(l,ik+)1) ⊂EW(Px,i,Z/p,lk)(lk) = Px(l,ik).
EW(P ,Z/p,l )
i k
99 ii❘❘❘❘❘❘❘❘❘❘❘❘❘❘6 V
f¯i ω EW(Px,i,Z/p,lk)
66 YY✹
✹
✹
✹
✹
✹
Pi+1cc●●●●●●●●●1 Q gii+1 //P(cid:15)(cid:15)igiii❙+❙1f¯❙|i❙|❙❙❙❙❙❙❙❙❙❙❙❙❙❙6 V (cid:15)(cid:15)ω| ✹✹✹✹✹✹✹✹✹λ✹✹
Px,i+1ff◆◆◆◆◆◆ν◆◆◆◆◆4 T i //Px,i ff◆◆◆◆◆◆◆◆◆◆✹◆✹✹◆✹◆✹✹4✹T✹, L
P(lk) f¯i| //--P(lk)
x,OOi+1 gi+1| x,i
i
µ
(cid:31)?
∂σ
10 L.Rubin,V.Toni´c
SinceLemma2.3istruefor|M| = P andn =l ,wehavef¯| ◦ν◦µ = λ◦f¯| ◦µ
x,i+1 k i Px,i+1 i P(lk)
x,i+1
is null-homotopic, and
r
H (f¯| ◦µ)(g ) = ε h ∈ H (P(lk)), where ε ≡ 0 (mod p).
lk i P(lk) σ e e lk x,i e
x,i+1 Xe=1
By Lemma 2.4 applied to P with n = l , we can select σ ,...,σ to be some (l +1)-
x,i+1 k 1 s k
simplexes of P so that the images g ,...,g of the generators of H (∂σ ),...,H (∂σ )
x,i+1 1 s lk 1 lk s
induced by the inclusions ∂σ ֒→ P(lk) form a basis for H (P(lk) ).
j x,i+1 lk x,i+1
Then for any g ∈ H (P(lk) ),
lk x,i+1
s s r
H (f¯| )(g) = H (f¯| )( m g ) = m ( ε h ),
lk i P(lk) lk i P(lk) j j j j,e e
x,i+1 x,i+1 Xj=1 Xj=1 Xe=1
where m ∈ Z, and ε ≡ 0 (mod p).
j j,e
Finally, since we know that gi+1| ≃ ω ◦ f¯| and ω| = id, we have that
i Px,i+1 i Px,i+1 P(lk)
x,i
gi+1| ≃ f¯| . Therefore H (gi+1| ) = H (f¯| ), so the last equation
i P(lk) i P(lk) lk i P(lk) lk i P(lk)
x,i+1 x,i+1 x,i+1 x,i+1
implies that H (gi+1| ;Z/p) is the zero-homomorphism. (cid:3)
lk i P(lk)
x,i+1
3. Proof of Theorem 1.1
Proof of Theorem 1.1: Choose a function ν :N → N such that for each i ∈ N,
(i) ν(i) ≤ i, and
(ii) ν−1(i) is infinite.
One may assume that X ⊂ Q = Hilbert cube. We are going to prove the existence for each
k ∈ N∪{∞} of a certain sequence S = (n , (Pk), ε , δ , (τk)), (τk)), j ∈ N, of entities,
j j j j j j j
and a sequence of maps (gj+1), j ∈ N, such that:
j e
• n ∈ N;
j
• P1 ⊂ P2 ⊂ ··· ⊂ P∞ are compact subpolyhedra of Inj;
j j j
• ε , δ > 0;
j j
• τ∞ is a triangulation of P∞ , τ∞ is a subdivision of τ∞,
j j j j
τk = τ∞| is a triangulation of Pk , τk = τ∞| is a subdivision of τk,
ej j Pjk j j j Pjk e j
(we will consider P∞ =(P∞,τ∞) and Pk = (P∞,τk));
e e j j j j j j e
• gj+1: P∞ → P∞ is a simplicial map relative to τ∞ and τ∞.
j j+1 j j+1 j