Chapter 3 – groups, trees and branched space boundaries
1 Introduction.
From the triangulation of the square as a 2-branched space we allocate a description
of its simplical homology. Extending this simplicial decomposition to n-branched
spaces, we are able to obtain an isomorphism with a construction on trees, thereby a
definition of the boundary of an n-branched space, and the corresponding simplicial
homology theory.
A branched space may be obtained via instances of trees, and trees with a standard
number of branches are related to free groups, where a regularly reconnected tree
provides a description of an arbitrary group with generators. This formulation allows
directly the introduction of a boundary as a terminal node.
Further, we introduce copies of trees and glue the nodes of each tree together, in a
process known as amalgamation. This leads directly to the description of branched
spaces. We can then implement the Poincaré characteristic of a branched space.
In chapter 1 we discussed the situation that  = 0 can fail for k-explosions. For the
case of groups we show in general that   0, as is evident in the classical treatment
of extended exact sequences for Ext and Tor.
Employing the concept of ladder numbers from chapter 2, we extend the group theory
to reconnected explosions, including in the transfinite case.
2 The theory for which branched spaces are a member.
The overarching theory we will adopt differs from topos theory. Like topos theory, we
can deal with morphisms, but since the exponential operator satisfies
(a ↑ b) ↑ c  a ↑ (b ↑ c)
in general, the morphisms we wish to deal with are not always associative.
It is sometimes stated (Bernstein, Matrices, Jacobsen, Algebra, Lawvere, Conceptual
mathematics, Rotman, Groups) that functions must be associative, but this is not the
case. If f, g and h are each octonionic multiplicative maps operating on a set of
octonions, then in general
f(gh)  (fg)h.
A topos has subobject classifiers. An example of this sort is shown in the Venn
diagram below for 2-subobject classifiers. We have two types of sets, A and the
complement of A in the set universe U.
U
A
1
An example of 3-subobject classifiers can be found in quantum chromodynamics.
There are three types of quark: red, green and blue. We retain the idea of n-subobject
classifiers, where n is a ladder number.
A topos has all finite limits. A reformulation of this is that it forms a Cartesian closed
category. An example is where all objects can be defined within a finite Cartesian
product A  B  ...  D. Under one interpretation we have introduced in the previous
chapter, ladder numbers do not have all finite limits. We can introduce sets of ladder
numbers (A a⇑ B) b⇑ ... c⇑ D ...) where a, b, c, d ... are themselves ladder numbers.
This absence of finite limits is a result of the fact that in this interpretation ladder
numbers are not well-ordered, but we may retain if we wish the axiom of choice
[Ad13], [Co62]. However, under term-by-term comparison, we can reintroduce wellordering, and thereby the topos property of having all finite limits.
Toposes have power objects, which we adopt here, but unlike the cardinal M we have
described in chapter 2, this axiom differs from what we have introduced in [Ad13]
concerning the absence of a distinct power object, and our ideas there on the
continuum hypothesis are in this sense pre-Cantorian.
3 Groups and trees.
Theorem. Any element of a finite group may be represented by a permutation. In
particular the element may be represented by a combination of two generators, a
cycle of n elements of the group and a swap transformation of two of them.
Proof. Provide n elements with a linear order over a cycle beginning at 1. Let a swap
transformation be a swap of two adjacent elements in this row of elements. We will
say the distance between these elements is 1. Then applying the swap again, the
identity transformation is obtained, and we will say the distance is 0. On applying one
move of the cycle and a swap, the distance between swapped elements increases to 2.
In similar manner, an arbitrary distance < n may be generated, and the start position of
the swap may be shifted by applying a cycle. 
For an infinite group the cyclic permutation is replaced by a shift, and the inverse shift
must be available. 
We can represent a group on n generators by a tree. For example suppose we have
a
b-1,
b
1
a-1
then we can represent a node at a by a continuation branch with ab, aa and ab-1, but do
not allow aa-1 since it is there already as 1 at the centre. Then each node is a distinct
element of a free group.
A reconnection of a tree is an identification of at least two of its nodes.
2
If the tree represents a group, if A and B represent two reconnected nodes, we say A-1
and B-1 also become reconnected.
If we have m trees (possibly m copies of the same tree), the m identifications of a
node from each tree is called an amalgamation.
When the trees represent groups, if A and B are two nodes which are amalgamated,
we will specify that A-1 and B-1 are also.
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3
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