Infinite Homotopy Theory

The homology of a locally finite graph with ends

“…The following lemma provides an easier way to do this, in terms of only the finite restrictions of the two words: (2). Hence by the infinity lemma there exists a sequence ' 0 ✓ ' 1 ✓ .…”

Brown representability for exterior cohomology and cohomology with compact supports

“…It is slightly more general than the original one introduced in [14], which include for instance, the classical CW-complexes (with topology as externology) and, in the proper case, the spherical objects under a tree of [2, §2], [3,IV] and [4, 1.3].…”

Uniformly continuous maps between ends of $${\mathbb{R}}$$ -trees

“…Consider f : S 1 → S 2 and g : S 2 → S 1 proper maps and proper homotopies H 1 of g • f to I d S 1 and H 2 of f • g to I d S 2 . First we construct two rooted proper maps 'near' f and g. Consider the unique arc in S 2 [x 2 , f (x 1 )] and, in order to define the rooted proper map from S 1 to S 2 , we are going to send this arc with a proper homotopy to the root x 2 and to pull somehow the rest of the tree after it.…”

“…It is well known, see for example 9.20 in [1], that since S i are locally finite simplicial trees, end(T 1 , v) = Fr(S 1 , v) = Fr(S 1 ) and end(T 2 , w) = Fr(S 2, w) = Fr(S 2 ) and, by 7.7, end(T 1 , v) ∼ = end(T 2 , w) ⇔ (T 1 , v) Mp (T 2 , w). If the metric is proper (T 1 , v) Mp (T 2 , w) ⇔ (T 1 , v) R P (T 2 , w) and hence Fr(S 1 ) ∼ = Fr(S 2 ) ⇔ (T 1 , v) R P (T 2 , w).…”

“…We finish this paper recovering, as a consequence of our constructions, the classical relation between the proper homotopy type of a locally finite simplicial tree and the topological type of its Freudenthal end space, see [1].…”