1. FIBRATIONS WITH UNIQUE PATH LIFTING PROPERTY
(Strobl, July 2011)
Joint with G. Conner

2. Unique path lifting fibration (UPL) is a fibration 𝑝:𝐸→𝑋 s.t. for
paths 𝛼, 𝛼 ′ :𝐼→𝐸 𝑝𝛼=𝑝 𝛼 ′ and 𝛼 0 = 𝛼 ′ 0 imply 𝛼= 𝛼 ′ .
𝑝:𝐸→𝑋 is a UPL 𝐸 𝐼 →𝐸 𝐵 𝐼 is a homeomorphism
(Spanier) a fibration 𝑝:𝐸→𝑋 is a UPL the fibres admit only constant paths
composition and product of UPLs is a UPL
Given a family 𝑋  →𝑋 of UPLs over 𝑋, then its fibred product  𝑋  →𝑋 (path component of the usual pull-back) is a UPL over 𝑋.

3. fibred product of all coverings over 𝑋
(universal UPL for coverings - admits unique projection to every covering)
fibred product of all UPLs over 𝑋
(universal UPL over 𝑋 - ‘supreme UPL’)
Conjecture
We are going to describe

4. cover of 𝑋:
let 𝜋 1 𝑋;U ≤ 𝜋 1 𝑋 be generated by
loops 𝛾𝛼 𝛾 where 𝛼 is a loop in some
𝑈∈U and 𝛾 is a path from 𝑥 0 to 𝛼(0).
Theorem (Spanier)
𝐺 is a covering subgorup  𝐺 contains some 𝜋 1 𝑋;U
Corollary
𝑓:𝑋→𝐾semi-locally 1-connected  Ker 𝑓 # is a covering subgorup
Proof
Take cover 𝑈 of 𝐾 with 𝜋 1 𝑈→𝐾 trivial. Then 𝜋 1 𝑋; 𝑓 −1 (𝑈)  Ker 𝑓 #

5. numerable cover of 𝑋 partition of unity defines 𝑓:𝑋→ U
Lemma
Proof
Wlog: U minimal i.e. for every 𝑈 there is 𝑥 𝑈 ∈𝑈 s.t. 𝜌 𝑈 𝑥 𝑈 =1.
Whenever 𝑈∩𝑉≠∅ choose path between 𝑥 𝑈 and 𝑥 𝑉 and define
𝑔: U (1) →𝑋 s.t. 𝑋 U
U (1) 𝑓 𝑔 𝑖  𝑔
Therefore 𝑓 # surjective.
Clearly 𝜋 1 𝑋;2U ≤Ker 𝑓 # .
For the converse use trick about 2-set simple covers.

6. 𝑋 has the doubling refinement property if every cover V has a refinement of the form 2U V. E.g. paracompact spaces.
Theorem
For 𝑋 with doubling refinement property
the following subgroups of 𝜋 1 𝑋 coincide:
intersection of all covering subgroups
intersection of all Ker 𝑓 # : 𝜋 1 𝑋 → 𝜋 1 𝐾 for 𝐾 polyhedron
shape kernel of 𝑋, ShKer 𝑋 = Ker 𝜋 1 𝑋 → 𝜋 1 𝑋
intersection of all 𝜋 1 𝑋;2U
Proof
(1) (2) because Ker 𝑓 # are covering subgroups
(2) (4) because (2) is contained in the intersection of all 𝜋 1 𝑋;2U
(by lemma) and the latter is contained in (4) by the doubling
refinement property
(4) (1) by Spanier’s theorem on covering groups
(2) = (3) standard

7. Theorem 𝑋 as before 
Proof 𝛼
𝛼 𝑖