Approximate fibrations on topological manifolds.

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“…These generalize unstratified results of Chapman [4], Hughes [10], and Hughes-TaylorWilliams [13]. Stratified sucking gives a condition for a map to be close to a manifold stratified approximate fibration, whereas stratified straightening can be thought of as a uniqueness result which gives a condition for two manifold stratified approximate fibrations to be controlled homeomorphic.…”

Controlled homotopy topological structures

“…The definition ofS^(p:E-^B) given above differs slightly from the definition given in [Hi,§8]. The two different complexes can be shown to be homotopy equivalent using the methods in [H 3 , §3].…”

“…However, we will not use that equivalence here. Instead, we will sketch a proof of an analogue of [Hi,Theorem 8.1].…”

“…{p : E -• B) is an n-parameter family of controlled structures on p : E -> B. The study of the homotopy relation in <9*(p : E -> B) was initiated in [Hi,§8]. …”

“…Unfortunately (p xid)/ need not be an approximate fibration. However, one can use [Hi,Theorem 5.1] to find an f.p. map / : …”