Hence, the associativity condition for the path product is not in general fulfilled.
However, in all instances, the paths p A = (p 1 p 2 ) p 3 and p B = p 1 (p 2 p 3 ) are
homotopically equivalent. Consequently, for the products of loop homotopy
equivalence classes, the associativity condition applies:
p 1
½ p 2
½
ð
Þp 3
½ ¼ p 1
½ p 2
½ p 3
½
ð
Þ :
Since all four conditions required for a group hold, these energy constrained
homotopy equivalence classes of loop paths form a group, denoted by π 1 (F(A), K 0 ).
In summary, we have the necessary group properties:
1. Closure property: each pair of homotopy classes have a product defined for them
that is also a homotpy class
2. There exists a unique unit element, [K 0 ].
3. Each homotopy equivalence class [p] has a unique inverse [p]
−1
4. The associativity condition holds, ([p 1 ][p 2 ]) [p 3 ] = [p 1 ] ([p 2 ][p 3 ]).
This group π 1 (F(A), K 0 ) is of relevance to the reaction mechanism problem on
F(A), however, as it stands, it has some apparent shortcomings. We can show,
though, that these shortcomings are of no significance, and, indeed, the group so
derived describes the most essential algebraic structure of all reaction mechanisms
on E(K), subject to the energy bound A.
Specifically, for the given energy constraint expressed by the level set F(A), this
group π 1 (F(A), K 0 ) appears highly restricted in one aspect: group π 1 (F(A), K 0 )
refers to a specific point K 0 and the associated unit element [K 0 ]. However, it can be
easily shown, that this group π 1 (F(A), K 0 ) is isomorphic with any other analogous
group π 1 (F(A), K′ 0 ) using a different reference point K′ 0 and the associated unit
element [K′ 0 ].
This can be demonstrated as follows. By considering any path p 00′ (u) connecting point K 0 to point K′ 0 , every loop path p(u) with origin at K 0 can be extended by
the two paths, p 00′ (u) and p 00′
−1 (u) into a loop path q with the new origin K′ 0
q ¼ p
À1
00 0 p p 00 0
For each choice of p, the two loops, p and q are clearly continuously deformable
into one another within F(A), in fact, the continuous deformation may occur within
the very point set represented by the path p 00′ (u), hence p and q are necessarily
members of the same homotopy equivalence class. That is, as abstract group, the
group π 1 (F(A), K 0 ), formed by homotopy equivalence classes with unit element
[K 0 ] is isomorphic with the group π 1 (F(A), K′ 0 ) of homotopy equivalence classes
with unit element [K′ 0 ].
That is, there is only one such abstract group, and its algebraic structure is
independent of the choice of the actual realization of the unit element [K 0 ], hence,
for this abstract group, the reference to any specific point K 0 can be omitted, and
one may simply write π 1 (F(A)).
250
P.G. Mezey
However, in all instances, the paths p A = (p 1 p 2 ) p 3 and p B = p 1 (p 2 p 3 ) are
homotopically equivalent. Consequently, for the products of loop homotopy
equivalence classes, the associativity condition applies:
p 1
½ p 2
½
ð
Þp 3
½ ¼ p 1
½ p 2
½ p 3
½
ð
Þ :
Since all four conditions required for a group hold, these energy constrained
homotopy equivalence classes of loop paths form a group, denoted by π 1 (F(A), K 0 ).
In summary, we have the necessary group properties:
1. Closure property: each pair of homotopy classes have a product defined for them
that is also a homotpy class
2. There exists a unique unit element, [K 0 ].
3. Each homotopy equivalence class [p] has a unique inverse [p]
−1
4. The associativity condition holds, ([p 1 ][p 2 ]) [p 3 ] = [p 1 ] ([p 2 ][p 3 ]).
This group π 1 (F(A), K 0 ) is of relevance to the reaction mechanism problem on
F(A), however, as it stands, it has some apparent shortcomings. We can show,
though, that these shortcomings are of no significance, and, indeed, the group so
derived describes the most essential algebraic structure of all reaction mechanisms
on E(K), subject to the energy bound A.
Specifically, for the given energy constraint expressed by the level set F(A), this
group π 1 (F(A), K 0 ) appears highly restricted in one aspect: group π 1 (F(A), K 0 )
refers to a specific point K 0 and the associated unit element [K 0 ]. However, it can be
easily shown, that this group π 1 (F(A), K 0 ) is isomorphic with any other analogous
group π 1 (F(A), K′ 0 ) using a different reference point K′ 0 and the associated unit
element [K′ 0 ].
This can be demonstrated as follows. By considering any path p 00′ (u) connecting point K 0 to point K′ 0 , every loop path p(u) with origin at K 0 can be extended by
the two paths, p 00′ (u) and p 00′
−1 (u) into a loop path q with the new origin K′ 0
q ¼ p
À1
00 0 p p 00 0
For each choice of p, the two loops, p and q are clearly continuously deformable
into one another within F(A), in fact, the continuous deformation may occur within
the very point set represented by the path p 00′ (u), hence p and q are necessarily
members of the same homotopy equivalence class. That is, as abstract group, the
group π 1 (F(A), K 0 ), formed by homotopy equivalence classes with unit element
[K 0 ] is isomorphic with the group π 1 (F(A), K′ 0 ) of homotopy equivalence classes
with unit element [K′ 0 ].
That is, there is only one such abstract group, and its algebraic structure is
independent of the choice of the actual realization of the unit element [K 0 ], hence,
for this abstract group, the reference to any specific point K 0 can be omitted, and
one may simply write π 1 (F(A)).
250
P.G. Mezey