By Mladen Bestvina
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Additional resources for Characterizing K-Dimensional Universal Menger Compacta
2. R e m a r k . If Po is a handlebody decomposition of a manifold, then any two associated maps are nomotopic [Waj]. If P 2 is a ^-partition, then any two associated maps are /i-nomotopic. 3. Setting. ), Q0 is a partition on M0 such that all elements of Q0 underlie a subcomplex of Lj, Mf is a (k — l)-connected manifold, Q is a //-partition on M', T 0 : QQ —• Q is an injective function that preserves intersections. 4. Step 1. ] ^0 > Q0 _ > © (/(? o n M[i] = * is a sufficiently large index).
Flp/ + i n M | y ] = J ? np/+i n M y , / > y). establishes (/3y+i) and (7y+i). 5. 5. 4), we conclude that (HQ) is a true statement. Let i(r + 1) = *, Pr+l = R0, T = T°. , P r +i is a partition and by (/30) and (7 0 ), T is a one-to-one correspondence. 6. L e m m a . -,qt intersection. D qt is a convenient partition. fl T(qt) is a one-to-one correspondence. Proof. 19 (the convenience part is obvious from the definition). D Corollary. e. 2) is (A; — l)-connected. 4. Corollary. Any triangulated //-manifold is LCk~l. Proof. 5. P r o p o s i t i o n. Any triangulated //-manifold M is /^-dimensional. Proof. The restriction to M of the canonical retraction of M,-+i onto the dual ^-skeleton of L,- defines a small map M —* ^-polyhedron. By [H-W] dim M
Characterizing K-Dimensional Universal Menger Compacta by Mladen Bestvina
Corollary. e. 2) is (A; — l)-connected. 4. Corollary. Any triangulated //-manifold is LCk~l. Proof. 5. P r o p o s i t i o n. Any triangulated //-manifold M is /^-dimensional. Proof. The restriction to M of the canonical retraction of M,-+i onto the dual ^-skeleton of L,- defines a small map M —* ^-polyhedron. By [H-W] dim M