Def 1. A dendrite $X$ is a 1 dimensional retract of the closed unit disk. Equivalently, $X$ is a compact, locally path connected, and uniquely arcwise connected metric space.

Def 2. The path $\alpha :[0,1]\rightarrow l_{2}$ ``factors through a dendrite loop'' if there exists a dendrite $X$, a loop $\gamma :[0,1]\rightarrow X$ (with $\gamma (0)=\gamma (1)$), and a map $\pi :X\rightarrow l_{2}$ such that $\alpha =\pi \gamma .$

Question. Suppose both the path $\alpha :[0,1]\rightarrow l_{2}$ and also the cancancatanted path $\alpha \ast \beta :[0,1]\rightarrow l_{2}$ can be factored through dendrite loops. Can $\beta $ be factored through a dendrite loop?

The following comments are intended to absorb some of the difficulties, and point to where the remaining difficulties lie.

The image of $\alpha$ can be an arbitrary Peano continuum embedded in Hilbert space $l_{2}$. However, all the difficult issues are present in the special case of the unit disk or unit cube.

Declare $\alpha :[0,1]\rightarrow l_{2}$ ``trivial'' if $\alpha $ factors through a dendrite loop. An elementary argument shows if the trivial loops $\alpha _{n}$ $\rightarrow \alpha $ uniformly then $\alpha $ is trivial, i.e. the trivial loops in $l_{2}$ are closed in the uniform topology.

(Apply Ascoli's theorem and the fact that each trivial loop extends to a map of the unit disk, so that each component of each point preimage separates the disk into convex sets).

Declare the path $\alpha $ to be ``tree-trivial'' if $\alpha $ lifts to a loop in a tree (a simply connected finite graph). An elementary proof shows $\beta $ is tree trivial if each of $\alpha $ and $\alpha \ast \beta $ is tree trivial.

An elementary proof shows if $\alpha $ is trivial then there exist tree- trivial loops $\alpha _{n}\rightarrow \alpha $.

(Lift $\alpha $ to a dendrite $X$ (a natural inverse limit of trees under strong deformation retraction, and project the retractions into $X.$ We can even assume $X$ is an $R-$tree, i.e. a length space)).

Consequently the question at hand is equivalent to the following: Suppose $\alpha $ and $\alpha \ast \beta $ are trivial. Suppose $\alpha _{n}\rightarrow \alpha $ and $\alpha _{n}$ is tree trivial. Must there exist tree trivial loops $\alpha _{n} \ast \beta _{n}\rightarrow \alpha \ast \beta ?$

Finally, it may be helpful consider the contrapositive, which can ultimately be reduced to the following:

Suppose the path $\alpha$ is irreducible, i.e. no subloop of the path $\alpha $ lifts to a dendrite loop. Suppose $\beta $ lifts to a dendrite loop. Can $\alpha \ast \beta $ lift to a dendrite loop?

  • $\begingroup$ What is $l_2$? An infinite-dimensional separable Hilbert space? $\endgroup$
    – YCor
    Mar 30 '17 at 2:10
  • $\begingroup$ Yes. In fact the Hilbert cube will suffice as a target, the topological countable product of closed intervals. The difficulties of the original question manifest themselves when the image of a loop is not 1 dimensional, for example if we assume all loops under consideration have image in 3 dimensional Euclidean space. $\endgroup$
    – Paul Fabel
    Mar 30 '17 at 5:35

The answer is apparently no.

A counterexample can be constructed as follows.

Partition the lower half of the closed unit disk into 0 cells, 1 cells, and 2 cells, so that each cell touches the boundary, the 1-cells touch at exactly their endpoints, and the 2 cells touch the boundary in 3 places, and so that no 0-cell touches the lower semicircle.

Arrange also that the union of the 2-cells is dense in the horizontal, and that the 0-cells have full linear measure in the horizontal

Topologically, the quotient space is a dendrite, and the boundary of the half disk determines a loop in the quotient space.

Now glue, (to the lower half disk) an upper closed half disk fibred (up to topological conjugacy) by the obvious semicircles.

Generically, now we have obtained a partition of the closed unit disk into continua so that the resulting quotient space is 2 dimensional, and deleting the reverse pair from the upper semicircle leaves an injective loop which, in particular, cannot be lifted to a loop in a dendrite.

This unexpected answer was in fact hiding in plain sight. But it was easy to be tricked for 3 reasons:

1) We get a yes answer for loops in 1 dimensional spaces.

2) The property of `lifting to a dendrite loop' is closed in the uniform topology for loops in any Peano continuum (with help from Ascoli's Theorem).

3) Dendrites are similar to trees. Both are compact contractible locally contractible uniquely arcwise connected metric spaces.

But unlike a tree, endpoints of a dendrite are typically dense G-delta! This fact is the real reason a counterexample can be built.

More details: Before gluing, the left and right sides of the lower unit disk determine dendrites. The 1 dimensional part of a dendrite (the complement of the endpoints) is F-sigma, a direct metric limit of trees. But the two F sigma sets corresponding to the reverse pair (created after mating) will typically be disjoint. After deleting the newly created reverse pair, (in the new quotient space) we are left with a loop which is injective and which cannot be lifted to a loop in a dendrite.


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