Resolution Theorems Levin Resolution Theorem for Q V. Toni´ c
Resolution Theorems Levin Resolution Theorem for Q Theorem (M. Levin, 2005) Let n ∈ N ≥ 2 . Then for every compact metrizable space X with dim Q X ≤ n, there exists a compact metrizable space Z and a surjective map π : Z → X such that π is Q -acyclic, and dim Z ≤ n. V. Toni´ c
� � � � Resolution Theorems Levin Resolution Theorem for Q Theorem (M. Levin, 2005) Let n ∈ N ≥ 2 . Then for every compact metrizable space X with dim Q X ≤ n, there exists a compact metrizable space Z and a surjective map π : Z → X such that π is Q -acyclic, and dim Z ≤ n. dim Z ≤ n dim Z ≤ n dim Z ≤ n Z π Z / p − acyclic Q − acyclic cell − like dim Z / p X ≤ n dim Z X ≤ n dim Q X ≤ n X V. Toni´ c
� � � � Resolution Theorems Levin Resolution Theorem for Q Theorem (M. Levin, 2005) Let n ∈ N ≥ 2 . Then for every compact metrizable space X with dim Q X ≤ n, there exists a compact metrizable space Z and a surjective map π : Z → X such that π is Q -acyclic, and dim Z ≤ n. dim Z ≤ n dim Z ≤ n dim Z ≤ n Z π Z / p − acyclic Q − acyclic cell − like dim Z / p X ≤ n dim Z X ≤ n dim Q X ≤ n X This does not work for any abelian group G : if G = Z / p ∞ = { m n ∈ Q / Z : n = p k for some k ≥ 0 } (quasi-cyclic p -group), then dim Z � n , but dim Z ≤ n + 1. V. Toni´ c
Resolution Theorems Levin Resolution Theorem for any G Theorem (M. Levin, 2003) Let G be an abelian group, n ∈ N ≥ 2 . Then for every compact metrizable space X with dim G X ≤ n, there exists a compact metrizable space Z and a surjective map π : Z → X such that: (a) π is G-acyclic, (b) dim Z ≤ n + 1 , and (c) dim G Z ≤ n. V. Toni´ c
� � � Resolution Theorems Levin Resolution Theorem for any G Theorem (M. Levin, 2003) Let G be an abelian group, n ∈ N ≥ 2 . Then for every compact metrizable space X with dim G X ≤ n, there exists a compact metrizable space Z and a surjective map π : Z → X such that: (a) π is G-acyclic, (b) dim Z ≤ n + 1 , and (c) dim G Z ≤ n. dim Z ≤ n dim Z ≤ n + 1 , dim G Z ≤ n Z π Q − acyclic G − acyclic dim Q X ≤ n dim G X ≤ n X V. Toni´ c
Possible generalization V. Toni´ c
� � � � � � Possible generalization Z X � Z � � Z ′ π π | X ′ � � � X � � � Z 2 � � � · · · � � � Z m � � � · · · � � � Z Z 1 π | π | π | π X 1 � � � X 2 � � � · · · � � � X m � � � · · · � � � X V. Toni´ c
� � � � � � � Possible generalization Z π X � � � Z Z ′ π | π X ′ � � � X � Z 2 � · · · � � � Z m � · · · � Z � � � � � � � � Z 1 π | π | π | π � X 2 � � � · · · � � � X m � � � · · · � X X 1 � � � � V. Toni´ c
� � � � � Possible generalization Z π X Z ′ Z X ′ � � � X � Z 2 � · · · � � � Z m � · · · � Z � � � � � � � � Z 1 π | π | π | π � X 2 � � � · · · � � � X m � � � · · · � X X 1 � � � � V. Toni´ c
� � � � � � � Possible generalization Z π X � Z � � Z ′ π π | X ′ � � � X � Z 2 � · · · � � � Z m � · · · � Z � � � � � � � � Z 1 π | π | π | π � X 2 � � � · · · � � � X m � � � · · · � X X 1 � � � � V. Toni´ c
� � � Possible generalization Z π X � � � Z Z ′ π | π � X X ′ � � · · · · · · Z 1 Z 2 Z m Z X 1 � � � X 2 � � � · · · � � � X m � � � · · · � � � X V. Toni´ c
� � � � � � � Possible generalization Z π X � � � Z Z ′ π | π � X X ′ � � � � � Z 2 � � � · · · � � � Z m � � � · · · � � � Z Z 1 π | π | π | π X 1 � � � X 2 � � � · · · � � � X m � � � · · · � � � X V. Toni´ c
Possible generalization enez-Rubin Theorem for Z Ageev-Jim´ dim Z 1 ≤ 1 dim Z 2 ≤ 2 dim Zk ≤ k Z 1 Z 2 Z k Z · · · · · · X 1 � � � X 2 � � � · · · � � � X k � � � � � X � · · · dim Z X 1 ≤ 1 dim Z X 2 ≤ 2 dim Z Xk ≤ k Theorem (S. Ageev, R. Jim´ enez and L. Rubin, 2004) Let X be a nonempty compact metrizable space and let X 1 ⊂ X 2 ⊂ . . . be a sequence of nonempty closed subspaces such that ∀ k ∈ N , dim Z X k ≤ k < ∞ . Then there exists a compact metrizable space Z, having closed subspaces Z 1 ⊂ Z 2 ⊂ . . . , and a (surjective) cell-like map π : Z → X, s.t. ∀ k ∈ N , (a) dim Z k ≤ k, (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a cell-like map. V. Toni´ c
� � � � Possible generalization enez-Rubin Theorem for Z Ageev-Jim´ dim Z 1 ≤ 1 dim Z 2 ≤ 2 dim Zk ≤ k � � � � � · · · � � � � � � � Z 2 � Z k � Z � · · · Z 1 π cell − like π | cell − like π | cell − like π | cell − like X 1 � � � X 2 � � � · · · � � � X k � � � � � X � · · · dim Z X 1 ≤ 1 dim Z X 2 ≤ 2 dim Z Xk ≤ k Theorem (S. Ageev, R. Jim´ enez and L. Rubin, 2004) Let X be a nonempty compact metrizable space and let X 1 ⊂ X 2 ⊂ . . . be a sequence of nonempty closed subspaces such that ∀ k ∈ N , dim Z X k ≤ k < ∞ . Then there exists a compact metrizable space Z, having closed subspaces Z 1 ⊂ Z 2 ⊂ . . . , and a (surjective) cell-like map π : Z → X, s.t. ∀ k ∈ N , (a) dim Z k ≤ k, (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a cell-like map. V. Toni´ c
Possible generalization Rubin-T. Theorem for Z / p dim Z 1 ≤ ℓ 1 dim Z 2 ≤ ℓ 2 dim Zk ≤ ℓ k Z 1 Z 2 Z k Z · · · · · · X 1 � � � X 2 � � � · · · � � � X k � � � � � X � · · · dim Z / p X 1 ≤ ℓ 1 dim Z / p X 2 ≤ ℓ 2 dim Z / p Xk ≤ ℓ k Theorem (L. Rubin and V. T., 2010) Let X be a nonempty compact metrizable space, let ℓ 1 ≤ ℓ 2 ≤ . . . be a sequence of natural numbers, and let X 1 ⊂ X 2 ⊂ . . . be a sequence of nonempty closed subspaces of X such that ∀ k ∈ N , dim Z / p X k ≤ ℓ k < ∞ . Then there exists a compact metrizable space Z, having closed subspaces Z 1 ⊂ Z 2 ⊂ . . . , and a (surjective) cell-like map π : Z → X, such that for each k in N , (a) dim Z k ≤ ℓ k , (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a Z / p-acyclic map. V. Toni´ c
� � � � Possible generalization Rubin-T. Theorem for Z / p dim Z 1 ≤ ℓ 1 dim Z 2 ≤ ℓ 2 dim Zk ≤ ℓ k � � � � � · · · � � � � � � � Z 2 � Z k � Z Z 1 � · · · π | Z / p − acyclic π | Z / p − acyclic π | Z / p − acyclic π cell − like X 1 � � � X 2 � � � · · · � � � X k � � � � � X � · · · dim Z / p X 1 ≤ ℓ 1 dim Z / p X 2 ≤ ℓ 2 dim Z / p Xk ≤ ℓ k Theorem (L. Rubin and V. T., 2010) Let X be a nonempty compact metrizable space, let ℓ 1 ≤ ℓ 2 ≤ . . . be a sequence of natural numbers, and let X 1 ⊂ X 2 ⊂ . . . be a sequence of nonempty closed subspaces of X such that ∀ k ∈ N , dim Z / p X k ≤ ℓ k < ∞ . Then there exists a compact metrizable space Z, having closed subspaces Z 1 ⊂ Z 2 ⊂ . . . , and a (surjective) cell-like map π : Z → X, such that for each k in N , (a) dim Z k ≤ ℓ k , (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a Z / p-acyclic map. V. Toni´ c
Possible generalization Conjecture for any abelian group G dim Z 1 ≤ ℓ 1+1 dim Z 2 ≤ ℓ 2+1 dim Zk ≤ ℓ k +1 dim G Z 1 ≤ ℓ 1 dim G Z 2 ≤ ℓ 2 dim G Zk ≤ ℓ k Z 1 Z 2 · · · Z k · · · Z X 1 � � � X 2 � � � · · · � � � X k � � � · · · � � � X dim G X 1 ≤ ℓ 1 dim G X 2 ≤ ℓ 2 dim G Xk ≤ ℓ k V. Toni´ c
Possible generalization Conjecture for any abelian group G dim Z 1 ≤ ℓ 1+1 dim Z 2 ≤ ℓ 2+1 dim Zk ≤ ℓ k +1 dim G Z 1 ≤ ℓ 1 dim G Z 2 ≤ ℓ 2 dim G Zk ≤ ℓ k Z 1 Z 2 · · · Z k · · · Z X 1 � � � X 2 � � � · · · � � � X k � � � · · · � � � X dim G X 1 ≤ ℓ 1 dim G X 2 ≤ ℓ 2 dim G Xk ≤ ℓ k where ℓ 1 ≤ ℓ 2 ≤ · · · ≤ ℓ k ≤ . . . is a sequence of numbers in N ≥ 2 . This would be a generalization of Levin’s theorem for any abelian group G . V. Toni´ c
� � � � Possible generalization Conjecture for any abelian group G dim Z 1 ≤ ℓ 1+1 dim Z 2 ≤ ℓ 2+1 dim Zk ≤ ℓ k +1 dim G Z 1 ≤ ℓ 1 dim G Z 2 ≤ ℓ 2 dim G Zk ≤ ℓ k � Z 2 � · · · � � � Z k � · · · � Z � � � � � � � � Z 1 π cell − like π | G − acyclic π | G − acyclic π | G − acyclic X 1 � � � X 2 � � � · · · � � � X k � � � · · · � � � X dim G X 1 ≤ ℓ 1 dim G X 2 ≤ ℓ 2 dim G Xk ≤ ℓ k where ℓ 1 ≤ ℓ 2 ≤ · · · ≤ ℓ k ≤ . . . is a sequence of numbers in N ≥ 2 . V. Toni´ c
� � � � Possible generalization Conjecture for any abelian group G dim Z 1 ≤ ℓ 1+1 dim Z 2 ≤ ℓ 2+1 dim Zk ≤ ℓ k +1 dim G Z 1 ≤ ℓ 1 dim G Z 2 ≤ ℓ 2 dim G Zk ≤ ℓ k � Z 2 � · · · � � � Z k � · · · � Z � � � � � � � � Z 1 π cell − like π | G − acyclic π | G − acyclic π | G − acyclic X 1 � � � X 2 � � � · · · � � � X k � � � · · · � � � X dim G X 1 ≤ ℓ 1 dim G X 2 ≤ ℓ 2 dim G Xk ≤ ℓ k where ℓ 1 ≤ ℓ 2 ≤ · · · ≤ ℓ k ≤ . . . is a sequence of numbers in N ≥ 2 . This would be a generalization of Levin’s theorem for any abelian group G . V. Toni´ c
Techniques used in proofs of resolution theorems V. Toni´ c
Techniques used in proofs of resolution theorems General idea for proving resolution theorems: V. Toni´ c
Techniques used in proofs of resolution theorems General idea for proving resolution theorems: Let X be a compact metrizable space with the required property (eg. dim G X ≤ n ). V. Toni´ c
Techniques used in proofs of resolution theorems General idea for proving resolution theorems: Let X be a compact metrizable space with the required property (eg. dim G X ≤ n ). Theorem (H. Freudenthal, 1937) Every compact metrizable space can be represented as the inverse limit of an inverse sequence of compact polyhedra, with surjective and simplicial bonding maps. V. Toni´ c
� � � � � Techniques used in proofs of resolution theorems General idea for proving resolution theorems: Let X be a compact metrizable space with the required property (eg. dim G X ≤ n ). Theorem (H. Freudenthal, 1937) Every compact metrizable space can be represented as the inverse limit of an inverse sequence of compact polyhedra, with surjective and simplicial bonding maps. · · · P i +1 · · · P 1 P 2 P i X f 2 f 3 f i +1 f i 1 2 i − 1 i (1) Choose an inverse sequence ( P i , f i +1 ) of compact polyhedra, i with simplicial, surjective bonding maps, whose limit is X . V. Toni´ c
� � � � � Techniques used in proofs of resolution theorems M 1 M 2 · · · M i M i +1 Z P 1 P 2 · · · P i P i +1 · · · X f 2 f 3 f i f i +1 1 2 i − 1 i (2) Use this sequence as a foundation to build another inverse sequence ( M i , g i +1 ) and an almost commutative ladder of i maps, so that lim( M i , g i +1 ) = Z and the map π : Z → X i with desired properties can be produced. V. Toni´ c
� � � � � � � � � � Techniques used in proofs of resolution theorems g i g i +1 g 2 g 3 i − 1 1 2 i M 1 M 2 · · · M i M i +1 · · · Z P 1 P 2 · · · P i P i +1 · · · X f 2 f 3 f i f i +1 1 2 i − 1 i (2) Use this sequence as a foundation to build another inverse sequence ( M i , g i +1 ) and an almost commutative ladder of i maps, so that lim( M i , g i +1 ) = Z and the map π : Z → X i with desired properties can be produced. V. Toni´ c
� � � � � � � � � � � � � � � Techniques used in proofs of resolution theorems g i g i +1 g 2 g 3 i − 1 1 2 i M 1 M 2 · · · M i M i +1 · · · Z φ 1 φ 2 φ i φ i +1 π P 1 P 2 · · · P i P i +1 · · · X f 2 f 3 f i f i +1 1 2 i − 1 i (2) Use this sequence as a foundation to build another inverse sequence ( M i , g i +1 ) and an almost commutative ladder of i maps, so that lim( M i , g i +1 ) = Z and the map π : Z → X i with desired properties can be produced. V. Toni´ c
� � � � � � Techniques used in proofs of resolution theorems · · · M i +1 M 1 M 2 M i Z φ 1 · · · P i +1 · · · P 1 P 2 P i X f 2 f 3 f i f i +1 1 2 i − 1 i The bottom inverse sequence is pre-chosen, while the top inverse sequence and the ladder of maps are built gradually. V. Toni´ c
� � � � � � � � Techniques used in proofs of resolution theorems g 2 1 M 1 M 2 · · · M i M i +1 Z φ 1 φ 2 P 1 P 2 · · · P i P i +1 · · · X f 2 f 3 f i f i +1 1 2 i − 1 i The bottom inverse sequence is pre-chosen, while the top inverse sequence and the ladder of maps are built gradually. V. Toni´ c
� � � � � � � � � � � Techniques used in proofs of resolution theorems g i g 2 g 3 i − 1 1 2 · · · M i +1 M 1 M 2 M i Z φ 1 φ 2 φ i · · · P i +1 · · · P 1 P 2 P i X f 2 f 3 f i f i +1 1 2 i − 1 i The bottom inverse sequence is pre-chosen, while the top inverse sequence and the ladder of maps are built gradually. V. Toni´ c
� � � � � � � � � � � � � Techniques used in proofs of resolution theorems g i g i +1 g 2 g 3 i − 1 1 2 i · · · M i +1 M 1 M 2 M i Z φ 1 φ 2 φ i φ i +1 · · · P i +1 · · · P 1 P 2 P i X f 2 f 3 f i f i +1 1 2 i − 1 i The bottom inverse sequence is pre-chosen, while the top inverse sequence and the ladder of maps are built gradually. V. Toni´ c
� � � � � � � � � � � � � � � Techniques used in proofs of resolution theorems g i g i +1 g 2 g 3 i − 1 1 2 i · · · M i +1 · · · M 1 M 2 M i Z φ 1 φ 2 φ i φ i +1 π · · · P i +1 · · · P 1 P 2 P i X f 2 f 3 f i f i +1 1 2 i − 1 i The bottom inverse sequence is pre-chosen, while the top inverse sequence and the ladder of maps are built gradually. V. Toni´ c
About the proof of Rubin-T. Theorem Part of the construction done in Hilbert cube I ℵ 0 = Q = I m × Q m ∞ � | x i − y i | with metric ρ ( x , y ) = . 2 i i =1 V. Toni´ c
About the proof of Rubin-T. Theorem Part of the construction done in Hilbert cube I ℵ 0 = Q = I m × Q m ∞ � | x i − y i | with metric ρ ( x , y ) = . (Hilbert cube is universal for 2 i i =1 metrizable compacta). V. Toni´ c
About the proof of Rubin-T. Theorem Part of the construction done in Hilbert cube I ℵ 0 = Q = I m × Q m ∞ � | x i − y i | with metric ρ ( x , y ) = . (Hilbert cube is universal for 2 i i =1 metrizable compacta). Maps p n i : Q → I n i are projections. V. Toni´ c
� � � � � � � � � � � � About the proof of Rubin-T. Theorem Part of the construction done in Hilbert cube I ℵ 0 = Q = I m × Q m ∞ � | x i − y i | with metric ρ ( x , y ) = . (Hilbert cube is universal for 2 i i =1 metrizable compacta). Maps p n i : Q → I n i are projections. We choose compact polyhedra . . . P 1 ⊂ I n 1 1 � � � � � � p n 1 p n 1 � � � � P 2 P 1 � � ⊂ I n 2 2 2 � � � � � � � � � p n 2 p n 2 p n 2 � � � � � � � P 2 � P 3 P 1 � � � � ⊂ I n 3 3 3 3 � � � � � � � � � � � � � p n 3 p n 3 p n 3 p n 3 � � � � � � � � � � � � . . . � � . . . . . . V. Toni´ c
� � � �� � �� � � � � � �� � � �� � � �� � � �� � � � About the proof of Rubin-T. Theorem . . . so that X = � ∞ i × Q n i , and X k = � ∞ i =1 P i i = k P k i × Q n i . P 1 1 × Q n 1 � � � � � � � � � � � � � � P 2 P 1 2 × Q n 2 2 × Q n 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 � P 3 P 1 3 × Q n 3 3 × Q n 3 3 × Q n 3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 � P 3 � P 4 P 1 4 × Q n 4 4 × Q n 4 4 × Q n 4 4 × Q n 4 � � � � � � � � � X X 1 X 2 X 3 · · · V. Toni´ c
�� �� � � � �� � � � � � � �� �� � � � � �� � � � � � About the proof of Rubin-T. Theorem Instead of the bottom inverse sequence we now have: P 1 1 × Q n 1 � � � � � � � � � � � � � � P 2 P 1 2 × Q n 2 2 × Q n 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 � P 3 P 1 3 × Q n 3 3 × Q n 3 3 × Q n 3 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 � P 3 � P 4 P 1 4 × Q n 4 4 × Q n 4 4 × Q n 4 4 × Q n 4 � � � � � � � � � X X 1 X 2 X 3 · · · V. Toni´ c
About the proof of Rubin-T. Theorem While choosing polyhedra, we simultaneously build simplicial maps g i +1 : P i +1 i +1 → P i i : i V. Toni´ c
� � � � � � � � � About the proof of Rubin-T. Theorem While choosing polyhedra, we simultaneously build simplicial maps g i +1 : P i +1 i +1 → P i i : i P 1 1 � � � � g 2 g 2 1 | 1 � � � � � � P 2 P 1 2 2 � � � � � � g 3 g 3 g 3 2 | 2 | 2 � � � � � � � � � � � P 2 � P 3 P 1 3 3 3 � � � � � � � � � � g 4 g 4 g 4 g 4 3 | 3 | 3 | 3 � � � � � � � � � � � � � � � � P 2 � P 3 � P 4 P 1 4 4 4 4 Z V. Toni´ c
About the proof of Rubin-T. Theorem i , g i +1 Our goal is: Z := lim( P i ) i V. Toni´ c
� � � � � � � � � About the proof of Rubin-T. Theorem i , g i +1 Our goal is: Z := lim( P i ) i P 1 1 � � � � g 2 g 2 1 | 1 � � � � � � P 2 P 1 2 2 � � � � � � g 3 g 3 g 3 2 | 2 | 2 � � � � � � � � � � � P 2 � P 3 P 1 3 3 3 � � � � � � � � � � g 4 g 4 g 4 g 4 3 | 3 | 3 | 3 � � � � � � � � � � � � � � � � P 2 � P 3 � P 4 P 1 4 4 4 4 Z V. Toni´ c
About the proof of Rubin-T. Theorem i ) ( ℓ k ) , g i +1 ...and Z k := lim(( P k ) i V. Toni´ c
� � � � � � � � � About the proof of Rubin-T. Theorem i ) ( ℓ k ) , g i +1 ...and Z k := lim(( P k ) i ( P 1 1 ) ( ℓ 1 ) � � � � g 2 g 2 1 | � 1 � � � � � � ( P 2 ( P 1 2 ) ( ℓ 1 ) 2 ) ( ℓ 2 ) � � � � � � � � g 3 g 3 g 3 2 | 2 | � � 2 � � � � � � � � � � � ( P 2 � ( P 3 ( P 1 3 ) ( ℓ 1 ) 3 ) ( ℓ 2 ) 3 ) ( ℓ 3 ) � � � � � � � � � � � � g 4 g 4 g 4 g 4 3 | 3 | 3 | � � � 3 � � � � � � � � � � � � � � � � ( P 2 � ( P 3 � ( P 4 ( P 1 4 ) ( ℓ 1 ) 4 ) ( ℓ 2 ) 4 ) ( ℓ 3 ) 4 ) ( ℓ 4 ) Z 1 Z 2 Z 3 Z 4 V. Toni´ c
About the proof of Rubin-T. Theorem To get the map π : Z → X , the following diagram should be very close to commuting: V. Toni´ c
� � � � � � � � � � � � � � � � � � � � � � About the proof of Rubin-T. Theorem To get the map π : Z → X , the following diagram should be very close to commuting: g 2 g 3 g 4 1 2 3 . . . P 1 P 2 P 3 Z 1 2 3 id id id p n 1 p n 2 p n 3 . . . P 1 P 2 P 3 π 1 2 3 . . . P 1 P 2 P 3 � � � � � � 1 × Q n 1 2 × Q n 2 3 × Q n 3 X V. Toni´ c
� � � � � � � � � � � � � � � � � � � � � � About the proof of Rubin-T. Theorem i and the maps g i +1 : P i +1 We choose both the polyhedra P i i +1 → P i i i as we go (the bottom sequence is not pre-chosen). g 2 g 3 g 4 1 2 3 . . . P 1 P 2 P 3 Z 1 2 3 id id id p n 1 p n 2 p n 3 . . . P 1 P 2 P 3 π 1 2 3 . . . P 1 P 2 P 3 1 × Q n 1 � � 2 × Q n 2 � � 3 × Q n 3 � � X V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes The hardest part of the construction is producing suitable g i +1 : P i +1 i +1 → P i i . i V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes The hardest part of the construction is producing suitable g i +1 : P i +1 i +1 → P i i . We use factoring maps through certain i CW-complexes – Edwards-Walsh complexes. V. Toni´ c
� About the proof of Rubin-T. Theorem Edwards-Walsh complexes The hardest part of the construction is producing suitable g i +1 : P i +1 i +1 → P i i . We use factoring maps through certain i CW-complexes – Edwards-Walsh complexes. EW ( L , G , n ) ω | L | For G an abelian group, n ∈ N and L a simplicial complex, an Edwards-Walsh resolution of L in dimension n is a pair ( 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, for each subcomplex L ′ of L , ω − 1 ( | L ′ | ) is a subcomplex of EW ( L , G , n )) such that: V. Toni´ c
� About the proof of Rubin-T. Theorem Edwards-Walsh complexes ω − 1 ( | L ′ | ) EW ( L , G , n ) ω | L ′ | | L | K ( G , n ) (i) ω − 1 ( | L ( n ) | ) = | L ( n ) | and ω | | L ( n ) | is the identity map of | L ( n ) | onto itself, V. Toni´ c
� About the proof of Rubin-T. Theorem Edwards-Walsh complexes ω − 1 ( | L ′ | ) EW ( L , G , n ) ω | L ′ | | L | K ( G , n ) (i) ω − 1 ( | L ( n ) | ) = | L ( n ) | and ω | | L ( n ) | is the identity map of | L ( n ) | onto itself, (ii) for every simplex σ of L with dim σ > n , the preimage ω − 1 ( σ ) is an Eilenberg-MacLane complex of type ( � G , n ), where the sum here is finite, and V. Toni´ c
� About the proof of Rubin-T. Theorem Edwards-Walsh complexes ω − 1 ( | L ′ | ) EW ( L , G , n ) ω | L ′ | | L | K ( G , n ) (i) ω − 1 ( | L ( n ) | ) = | L ( n ) | and ω | | L ( n ) | is the identity map of | L ( n ) | onto itself, (ii) for every simplex σ of L with dim σ > n , the preimage ω − 1 ( σ ) is an Eilenberg-MacLane complex of type ( � G , n ), where the sum here is finite, and (iii) for every subcomplex L ′ of L and every map f : | L ′ | → K ( G , n ), the composition f ◦ ω | ω − 1 ( | L ′ | ) : ω − 1 ( | L ′ | ) → K ( G , n ) extends to a map F : EW ( L , G , n ) → K ( G , n ). V. Toni´ c
� � � � � About the proof of Rubin-T. Theorem Edwards-Walsh complexes ω − 1 ( | L ′ | ) EW ( L , G , n ) � � � � � � � F � � ω ω | � � � � � � � K ( G , n ) | L ′ | | L | � � f (i) ω − 1 ( | L ( n ) | ) = | L ( n ) | and ω | | L ( n ) | is the identity map of | L ( n ) | onto itself, (ii) for every simplex σ of L with dim σ > n , the preimage ω − 1 ( σ ) is an Eilenberg-MacLane complex of type ( � G , n ), where the sum here is finite, and (iii) for every subcomplex L ′ of L and every map f : | L ′ | → K ( G , n ), the composition f ◦ ω | ω − 1 ( | L ′ | ) : ω − 1 ( | L ′ | ) → K ( G , n ) extends to a map F : EW ( L , G , n ) → K ( G , n ). V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes Not all of the abelian groups G admit an Edwards-Walsh resolution (for any simplicial complex). V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes Not all of the abelian groups G admit an Edwards-Walsh resolution (for any simplicial complex). But when G is Z or Z / p , Edwards- Walsh resolutions exist for any simplicial complex L . In fact: V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes Not all of the abelian groups G admit an Edwards-Walsh resolution (for any simplicial complex). But when G is Z or Z / p , Edwards- Walsh resolutions exist for any simplicial complex L . In fact: Lemma 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 : 1 the ( n + 1) -skeleton of EW ( L , Z , n ) is equal to L ( n ) ; 2 the ( n + 1) -skeleton of EW ( L , Z / p , n ) is obtained from L ( n ) by attaching ( n + 1) -cells by a map of degree p to the boundary ∂σ , for every ( n + 1) -dimensional simplex σ . V. Toni´ c
About the proof of Rubin-T. Theorem Edwards-Walsh complexes Not all of the abelian groups G admit an Edwards-Walsh resolution (for any simplicial complex). But when G is Z or Z / p , Edwards- Walsh resolutions exist for any simplicial complex L . In fact: Lemma 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 : 1 the ( n + 1) -skeleton of EW ( L , Z , n ) is equal to L ( n ) ; 2 the ( n + 1) -skeleton of EW ( L , Z / p , n ) is obtained from L ( n ) by attaching ( n + 1) -cells by a map of degree p to the boundary ∂σ , for every ( n + 1) -dimensional simplex σ . Describe how to build an EW ( L , Z / p , n ). V. Toni´ c
� � About the proof of Rubin-T. Theorem Edwards-Walsh complexes Edwards-Walsh complexes (resolutions) are useful because Lemma Let X be a compact metrizable space with dim G X ≤ n, and let L be a finite simplicial complex. Then for every Edwards-Walsh resolution ω : EW ( L , G , n ) → | L | , and for every map f : X → | L | , there exists an approximate lift � f : X → EW ( L , G , n ) of f . EW ( L , G , n ) � f is an approximate lift of f w.r. � � � f � ω to ω if ∀ x ∈ X , f ( x ) ∈ ∆ ⇒ � � ω ◦ � � f ( x ) ∈ ∆. � | L | X f dim G X ≤ n ⇔ X τ K ( G , n ) V. Toni´ c
� � About the proof of Rubin-T. Theorem Edwards-Walsh complexes Edwards-Walsh complexes (resolutions) are useful because Lemma Let X be a compact metrizable space with dim G X ≤ n, and let L be a finite simplicial complex. Then for every Edwards-Walsh resolution ω : EW ( L , G , n ) → | L | , and for every map f : X → | L | , there exists an approximate lift � f : X → EW ( L , G , n ) of f . EW ( L , G , n ) � f is an approximate lift of f w.r. � � � f � ω to ω if ∀ x ∈ X , f ( x ) ∈ ∆ ⇒ � � ω ◦ � � f ( x ) ∈ ∆. � | L | X f dim G X ≤ n ⇔ X τ K ( G , n ) V. Toni´ c
About the proof of Rubin-T. Theorem Construction of polyhedra Construction is inductive. V. Toni´ c
� � � � � About the proof of Rubin-T. Theorem Construction of polyhedra Construction is inductive. Induction step: suppose we have built P 1 1 � � � g 2 g 2 1 | 1 � � � � � � P 2 P 1 2 2 � � � � � � � � g i g i g i i − 1 | i − 1 | i − 1 � � � � � � � � � . . . � � � P 2 � P i P 1 i i i . . . . . . P i +1 P 1 P 2 i +1 i +1 i +1 V. Toni´ c
� � � � � � � � � � About the proof of Rubin-T. Theorem Construction of polyhedra We would like to build: P 1 1 � � � g 2 g 2 1 | 1 � � � � � � P 2 P 1 2 2 � � � � � � � � g i g i g i i − 1 | i − 1 | i − 1 � � � � � � � � � . . . � � � P 2 � P i P 1 i i i � � � � � � g i +1 g i +1 g i +1 | | i i i � � � � � � i +1 � � i +1 � � . . . � � P 1 P 2 . . . � P i +1 i +1 V. Toni´ c
About the proof of Rubin-T. Theorem Construction of polyhedra To get Z / p -acyclicity of π | Z k : Z k → X k : V. Toni´ c
� � About the proof of Rubin-T. Theorem Construction of polyhedra To get Z / p -acyclicity of π | Z k : Z k → X k : . . . EW ( P k i , Z / p ,ℓ k ) � � � � � � � � � g i i − 1 | � � � P k i � � � � � � � � � g i +1 | � � i P k i +1 within each of our diagonals, we need to have that, for infinitely many indexes i , g i +1 | factors up to homotopy through an i Edwards-Walsh complex: V. Toni´ c
� � � � About the proof of Rubin-T. Theorem Construction of polyhedra To get Z / p -acyclicity of π | Z k : Z k → X k : . . . EW ( P k i , Z / p ,ℓ k ) � � � � � � � � � ω � � � � � g i i − 1 | � � � � � � � f P k � � i � � � � � � � � � � � � � � � � g i +1 | � � � i P k i +1 within each of our diagonals, we need to have that, for infinitely many indexes i , g i +1 | factors up to homotopy through an i Edwards-Walsh complex: g i +1 | ≃ ω ◦ � f . i V. Toni´ c
� � � � About the proof of Rubin-T. Theorem Construction of polyhedra To get Z / p -acyclicity of π | Z k : Z k → X k : . . . EW ( P k i , Z / p ,ℓ k ) � � � � � � � � � ω � � � � � g i � i − 1 | � � � � � � f P k � � i � � � � � � � � � � � � � � � � g i +1 | � � � i P k i +1 So we will have to choose a “book-keeping” function ν : N → N to tell us on which diagonal to focus next. V. Toni´ c
� � � � About the proof of Rubin-T. Theorem Construction of polyhedra To get Z / p -acyclicity of π | Z k : Z k → X k : . . . EW ( P k i , Z / p ,ℓ k ) � � � � � � � � � ω � � � � � g i � i − 1 | � � � � � � f P k � � i � � � � � � � � � � � � � � � � g i +1 | � � � i P k i +1 So we will have to choose a “book-keeping” function ν : N → N to tell us on which diagonal to focus next. ν ( i ) ≤ i , ν − 1 ( k ) is infinite. V. Toni´ c
� � � � � About the proof of Rubin-T. Theorem Let’s suppose our “book-keeping” function told us to focus on ν ( i ) = k ≤ i . This means: focus on X k and build EW ( P k i , Z / p , ℓ k ) above P k i . P 1 1 � � � g 2 g 2 1 | 1 � � � � � � P 2 EW ( P k P 1 i , Z / p , ℓ k ) 2 2 � � � � � � � � � � � � � � g i g i g i i − 1 | i − 1 | � i − 1 � � � � � � � � � � � � � � . . . � � � � � . . . � � � P k � P i P 1 i i i X k V. Toni´ c
� � � � � � About the proof of Rubin-T. Theorem Let’s suppose our “book-keeping” function told us to focus on ν ( i ) = k ≤ i . This means: focus on k -th diagonal and build EW ( P k i , Z / p , ℓ k ) above P k i . P 1 1 � � � g 2 g 2 1 | 1 � � � � � � P 2 EW ( P k P 1 i , Z / p , ℓ k ) 2 2 ω � � � � � � � � � � � � � � g i g i g i i − 1 | i − 1 | � i − 1 � � � � � � � � � � � � � � . . . � � � � � . . . � � � P k � P i P 1 i i i X k V. Toni´ c
� � � � � � � � About the proof of Rubin-T. Theorem Now there is an approximate lift f : X k → EW ( P k i , Z / p , ℓ k ) of p n i | : X k → P k i (because dim Z / p X k ≤ ℓ k ). P 1 1 � � � g 2 g 2 1 | 1 � � � � � � P 2 EW ( P k P 1 i , Z / p , ℓ k ) 2 2 � � � � ω � � � � � � � � � � � � � � � g i � g i g i f i − 1 | i − 1 | � � i − 1 � � � � � � � � � � � � � � . . . � � � � � . . . � � � � P k � P i P 1 i i i � � � � � � � � � � � � p ni | � � � � X k V. Toni´ c
� � � About the proof of Rubin-T. Theorem We can extend f over a nbhd U of X k in Hilbert cube Q , then make this nbhd smaller so that on U maps p n i and ω ◦ f are close. EW ( P k i , Z / p , ℓ k ) � � � ω � � � � � f � P k � � i � � � � � � � � � � � � � � � � p ni | � � � � � X k U V. Toni´ c
� � � � � About the proof of Rubin-T. Theorem We can extend f over a nbhd U of X k in Hilbert cube Q , then make this nbhd smaller so that on U maps p n i and ω ◦ f are close. EW ( P k i , Z / p , ℓ k ) � � � � � ω � � � � � � � � f f � P k � � � � i � � � � � � � � � � � � � � � � � � � p ni | � � � � � � � � � X k U V. Toni´ c
About the proof of Rubin-T. Theorem Now you can pick n i +1 , as well as the polyhedra P i +1 i +1 in I n i +1 so that they i +1 ⊃ P i i +1 ⊃ . . . ⊃ P k i +1 ⊃ . . . ⊃ P 1 satisfy a number of technical properties, including X k ⊂ P k i +1 × Q n i +1 ⊂ U . V. Toni´ c
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