arXiv:1101.1700v3 [math.CT] 4 Nov 2012

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE
KOUKI TANIYAMA
Abstract. We define a concept which we call multiplicity. First, multiplicity
of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from
one to another. Based on this multiplicity, we define a pseudo distance on the
class of objects. We define and study several multiplicities in the category of
topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated R-modules and R-linear maps over
a principal ideal domain R, and the neighbourhood category of oriented knots
in the 3-sphere.

1. Introduction
Let C be a category with objects X, Y, · · · . We denote the set of morphisms from
X to Y by Hom(X, Y ). By f : X → Y we mean f ∈ Hom(X, Y ). The composition
of f : X → Y and g : Y → Z is denoted by g ◦ f : X → Z. The identity morphism
on X is denoted by idX : X → X. Note that by the definition of category the
following (1), (2) and (3) hold.
(1) For any f : X → Y , g : Y → Z and h : Z → W , h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
(2) For any f : X → Y , f ◦ idX = f and for any g : Y → X, idX ◦ g = g.
(3) Hom(X, Y ) and Hom(Z, W ) are disjoint unless X = Z and Y = W .
Let R be the set of all real numbers and N the set of all natural numbers. For a
real number a we denote the set of all real numbers greater than or equal to a by
R≥a . Similarly we denote the set of all real numbers greater than a by R>a . Let ∞
be an element that is not in R. We extend the order, addition and multiplication
of R to R ∪ {∞} in the usual way. Namely, for any real number r it holds r ≤ ∞,
∞ ≤ ∞, r + ∞ = ∞ + r = ∞ + ∞ = ∞ and for any positive real number r it holds
r · ∞ = ∞ · r = ∞ · ∞ = ∞.
Suppose that for each morphism f : X → Y , an element m(f ) of R≥1 ∪ {∞} is
assigned such that the following (1) and (2) hold.
(1) m(idX ) = 1 for any object X.
(2) For any f : X → Y and g : Y → Z, m(g ◦ f ) ≤ m(f )m(g).
Then we say that m is a multiplicity on the category C. Let m(X : Y ) be the
infimum of m(f ) where f varies over all elements of Hom(X, Y ). If there are no
morphisms from X to Y then we define m(X : Y ) = ∞. We call m(X : Y ) the
multiplicity of X over Y .
Proposition 1.1. (1) For any objects X and Y , m(X : Y ) ≥ 1.
2000 Mathematics Subject Classification. Primary 18D99; Secondary 13C05, 20E99, 57M25,
57M99.
Key words and phrases. multiplicity, category, topological space, group, module, knot.
The author was partially supported by Grant-in-Aid for Scientific Research (C) (No. 21540000),
Japan Society for the Promotion of Science.
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KOUKI TANIYAMA

(2) For any object X, m(X : X) = 1.
(3) For any objects X, Y and Z,
m(X : Z) ≤ m(X : Y )m(Y : Z).
Proof. (1) By definition we have m(X : Y ) ≥ 1.
(2) We have m(X : X) = m(idX ) = 1.
(3) For any ε > 0 there exists f : X → Y with

√
ε
3ε
,
}
m(f ) − m(X : Y ) < min{
3m(Y : Z) 3
and g : Y → Z with
√
ε
3ε
m(g) − m(Y : Z) < min{
,
}.
3m(X : Y ) 3
Then we have
m(f )m(g) − m(X : Y )m(Y : Z)
= m(f )m(g) − m(f )m(Y : Z) + m(f )m(Y : Z) − m(X : Y )m(Y : Z)
= m(f )(m(g) − m(Y : Z)) + (m(f ) − m(X : Y ))m(Y : Z)
√
ε
3ε
< (m(X : Y ) +
)(m(g) − m(Y : Z)) +
m(Y : Z)
3
3m(Y : Z)
√ √
ε
ε
3ε 3ε
< m(X : Y )
+
+
m(Y : Z)
3m(X : Y )
3
3
3m(Y : Z)
= ε.
Since m(X : Z) ≤ m(g ◦ f ) ≤ m(f )m(g) we have m(X : Z) − m(X : Y )m(Y : Z) <
ε. Since ε is any positive number we have the conclusion. 
Remark 1.2. All examples of m in this paper take their values in a proper subset
N ∪ {∞} of R≥1 ∪ {∞} or a proper subset {en |n ∈ {0} ∪ N} of R≥1 ∪ {∞} where e is
the base of natural logarithm. Then m(X : Y ) is simply defined as the minimum,
not as the infimum, and the proof of Proposition 1.1 (3) becomes much simpler.
However we define the range of m as above for potential future use.
Let dm (X, Y ) be an element of R≥0 ∪ {∞} defined by the following.
dm (X, Y ) = loge (m(X : Y )m(Y : X)).

Here we define loge (∞) = ∞ as usual. We call dm (X, Y ) the multiplicity distance
of X and Y .
We say that X has finite multiplicity property over Y with respect to m if m(X :
Y ) 6= ∞, namely if there is a morphism f : X → Y with m(f ) < ∞. We say that
m has finite multiplicity property if any object X has finite multiplicity over any
object Y .
Proposition 1.3. Let m be a multiplicity on a category C that has finite multiplicity
property. Then dm is a pseudo distance on the class of objects of C. Namely the
following (D1′ ), (D2) and (D3) hold for any objects X, Y and Z of C.
(D1′ ) dm (X, Y ) ≥ 0, dm (X, X) = 0,
(D2) dm (X, Y ) = dm (Y, X),
(D3) dm (X, Z) ≤ dm (X, Y ) + dm (Y, Z).

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Proof. (D1′ ) By Proposition 1.1 (1) we have m(X : Y )m(Y : X) ≥ 1. Therefore
dm (X, Y ) = loge (m(X : Y )m(Y : X)) ≥ 0. By Proposition 1.1 (2) we have
m(X : X)m(X : X) = 1. Therefore dm (X, X) = loge (m(X : X)m(X : X)) = 0.
(D2) By definition we have dm (X, Y ) = dm (Y, X).
(D3) By Proposition 1.1 (3) we have m(X : Z)m(Z : X) ≤ m(X : Y )m(Y :
Z)m(Z : Y )m(Y : X) = m(X : Y )m(Y : X)m(Y : Z)m(Z : Y ). Therefore
dm (X, Z) = loge (m(X : Z)m(Z : X)) ≤ loge (m(X : Y )m(Y : X)m(Y : Z)m(Z :
Y )) = loge (m(X : Y )m(Y : X))+ loge (m(Y : Z)m(Z : Y )) = dm (X, Y )+ dm (Y, Z).

The following is a typical example of multiplicity. Let Cset be the category of
non-empty sets and maps. Then a morphism f : X → Y is a map from a set X
to a set Y . We denote the cardinality of a set A by |A|. For any infinite set A,
the cardinality of A is denoted by the same symbol |A| = ∞. Let mmap (f ) be an
element of N ∪ {∞} defined by
mmap (f ) = sup{|f −1 (y)| | y ∈ Y }.
We call mmap map-multiplicity.
Proposition 1.4. A map-multiplicity mmap is a multiplicity. Namely the following
(1) and (2) hold.
(1) mmap (idX ) = 1 for any set X of Cset .
(2) For any maps f : X → Y and g : Y → Z, mmap (g ◦ f ) ≤ mmap (f )mmap (g).
Proof. The proof of (1) is trivial. We will show (2). If mmap (f ) = ∞ or mmap (g) =
∞ then the inequality holds. Suppose mmap (f ) < ∞ and mmap (g) < ∞. Let z
be any element of Z. Since |g −1 (z)| ≤ mmap (g) and |f −1 (y)| ≤ mmap (f ) for any
y ∈ Y we have |(g ◦ f )−1 (z)| = | ∪y∈g−1 (z) f −1 (y)| ≤ mmap (f )mmap (g). Thus we
have mmap (g ◦ f ) ≤ mmap (f )mmap (g). 
The following proposition shows that map-multiplicity is, in a sense, generalization of division. For a real number a we denote the least integer no less than a by
⌈a⌉.
Proposition 1.5. Let X and Y be finite sets. Then the following holds.
mmap (X : Y ) = ⌈|X|/|Y |⌉.
Proof. For any map f : X → Y we have mmap (f ) ≥ ⌈|X|/|Y |⌉ by the pigeonhole
principle, and clearly, there is a map g : X → Y with mmap (g) = ⌈|X|/|Y |⌉.
Therefore we have the result. 
Let C1 and C2 be categories and m2 a multiplicity on C2 . Let F be a functor
from C1 to C2 . For each morphism f of C1 we define m1 (f ) by m1 (f ) = m2 (F (f )).
We call m1 the pull-back multiplicity of m2 with respect to a functor F .
Proposition 1.6. The pull-back multiplicity m1 is a multiplicity. Namely the
following (1) and (2) hold.
(1) m1 (idX ) = 1 for any object X of C1 .
(2) For any morphisms f : X → Y and g : Y → Z of C1 , m1 (g ◦ f ) ≤ m1 (f )m1 (g).

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Proof. (1) By definition we have m1 (idX ) = m2 (F (idX )). Since F is a functor we
have F (idX ) = idF (X) . Therefore m2 (F (idX )) = m2 (idF (X) ) = 1.
(2) By definition we have m1 (g ◦ f ) = m2 (F (g ◦ f )). Since F is a functor we
have F (g ◦ f ) = F (g) ◦ F (f ). Therefore m2 (F (g ◦ f )) = m2 (F (g) ◦ F (f )) ≤
m2 (F (f ))m2 (F (g)) = m1 (f )m1 (g). 
Suppose that a category has a functor to the category of non-empty sets and
maps, then the pull-back multiplicity of the map-multiplicity is also called the
map-multiplicity so long as no confusion occurs. Many categories, such as the
category of topological spaces and continuous maps and the category of groups and
homomorphisms, have a forgetful functor to the category of non-empty sets and
maps. Thus the map-multiplicity is defined on such categories. We will study them
in Section 2 and Section 3.
Let X and Y be objects. Suppose that there exists an injective morphism from
X to Y , and there exists an injective morphism from Y to X. Then we consider
the problem whether or not X and Y are isomorphic. We consider this problem
in several categories in the following sections. An analogous problem for surjective
morphism is also considered in Section 3 and Section 4.
Remark 1.7. In [4] a natural number b(f ) is defined for an embedding f : G → S3
of a finite graph G into the 3-sphere S3 as a generalization of the braid index of
knots and links. Then b(f ) is estimated below by mmap (g) for any continuous
map g : G → S1 from G to the unit circle S1 . Therefore the author defined
map-multiplicity of continuous maps and then generalized it to multiplicities in
various categories. There are some preliminary announcements of this work. See
[9], [10] and [11]. In [2] S. Bogatyi, J. Fricke and E. Kudryavtseva independently
defined the same number as our map-multiplicity of a continuous map, and studied
it in somewhat different interest. In [3] M. Gromov independently defined the
cardinality of a topological space over another topological space which is essentially
the same as our map-multiplicity of a topological space over another topological
space, and studied it in various aspects. In [5] R. Karasev also showed some results
on multiplicity of continuous maps between manifolds.
2. Multiplicity of topological spaces
Let X and Y be topological spaces and f : X → Y a continuous map from X to
Y . Then the map-multiplicity of f is defined by mmap (f ) = sup{|f −1 (y)| | y ∈ Y }
and the map-multiplicity mmap (X : Y ) of X over Y is defined to be the infimum
of mmap (f ) where f varies over all continuous maps from X to Y .
We say that topological spaces X and Y are weakly homeomorphic if there exists
a continuous injection from X to Y and there exists a continuous injection from Y
to X. We say that a set of topological spaces T is classed if any two weakly homeomorphic topological spaces in T are homeomorphic. We say that a topological space
X is self-closed if every continuous injection from X to X is a homeomorphism.
Proposition 2.1. A set of self-closed topological spaces is classed.
Proof. Let X and Y be self-closed topological spaces. Suppose that X and Y
are weakly homeomorphic. Then there are continuous injections f : X → Y and
g : Y → X. Then g ◦ f : X → X is a continuous injection. Since X is self-closed

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5

g ◦ f is a homeomorphism. Similarly f ◦ g : Y → Y is a homeomorphism. Therefore
both f and g are bijections. Since f −1 : Y → X is a composition of two continuous
maps g : Y → X and (g ◦ f )−1 : X → X, f −1 is also continuous. Therefore f is a
homeomorphism. 
Theorem 2.2. Let X be a classed set of topological spaces. Let CX be a category
whose objects are the elements of X and morphisms are the continuous maps between the elements of X . Let mmap be the map-multiplicity on CX and dmmap the
map-multiplicity distance. Let X and Y be elements of X . Then X and Y are
homeomorphic if and only if dmmap (X, Y ) = 0. Thus, if mmap has finite multiplicity property on X , then the pseudo distance dmmap defines a distance on the set of
homeomorphism classes of X .
Proof. Suppose that X and Y are homeomorphic. Then there is a homeomorphism
f : X → Y . Since mmap (f ) = mmap (f −1 ) = 1 we have mmap (X : Y ) = mmap (Y :
X) = 1. Therefore dmmap (X, Y ) = 0. Suppose dmmap (X, Y ) = 0. Then there are
continuous injections f : X → Y and g : Y → X. Therefore X and Y are weakly
homeomorphic. Since X is classed, X and Y are homeomorphic. 
In the following we consider simplicial complexes. All simplicial complexes in
this paper are locally finite. Note that a simplicial complex is formally a set of
simplexes. However we do not distinguish a simplicial complex and the union of its
simplices so long as no confusion occurs.
Lemma 2.3. Let G be a finite one-dimensional simplicial complex. Let X be a
subspace of G that is homeomorphic to a finite one-dimensional complex. Then the
following (1), (2) and (3) hold.
(1) For any 1-simplex e of G, e ∩ X is a disjoint union of finitely many points and
closed intervals.
(2) There is a subdivision G′ of G such that X is a sub-complex of G′ .
(3) The first Betti number β1 (X) of X is less than or equal to the first Betti number
β1 (G) of G.

Proof. (1) Since X is compact and G is Hausdorff, X is closed in G. Also e is
closed in G. Thus e ∩ X is a closed subset of a compact set G, hence compact. Each
connected component of e ∩ X is a compact connected subset of e. Therefore it is
a point or a closed interval. Suppose that the number of connected components of
e ∩ X is not finite. Since X is compact the number of connected components of X
is finite. Therefore there are connected components A1 , A2 and A3 of e ∩ X that
are contained in a connected component B of X. We may suppose without loss
of generality that A1 , A2 and A3 are arranged in this order in e. Then there are
points x and y of e \ X such that x is between A1 and A2 and y is between A2 and
A3 . But then A2 is separated from A1 and A3 in X and they cannot be contained
in the same connected component of X. This is a contradiction. Therefore the
number of connected components of e ∩ X is finite. Thus e ∩ X is a disjoint union
of finitely many points and closed intervals.
(2) By (1) we immediately have the result.

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KOUKI TANIYAMA

(3) By (2), X is a sub-complex of a finite one-dimensional simplicial complex G′ .
Then the inequality β1 (X) ≤ β1 (G′ ) is a well-known fact in combinatorial topology.
Since β1 (G′ ) = β1 (G) we have the result. 
Proposition 2.4. (1) A compact Hausdorff space X is self-closed if and only if no
proper subspace of X is homeomorphic to X.
(2) A closed manifold is self-closed.
(3) Let G be a finite one-dimensional simplicial complex. Suppose that no 0-simplex
is a face of exactly one 1-simplex. Then G is self-closed.

Proof. (1) Suppose that a proper subset Y of X is homeomorphic to X. Let
f : X → Y be a homeomorphism and i : Y → X the inclusion map. Then the
composition i ◦ f : X → X is a continuous injection. Since i ◦ f is not surjective
it is not a homeomorphism. Therefore X is not self-closed. Suppose that X is
not self-closed. Then there is a continuous injection f : X → X that is not a
homeomorphism. Since the source X is compact and the target X is Hausdorff,
f is a closed map. Therefore X and f (X) are homeomorphic. Since f is not a
homeomorphism, f (X) is a proper subspace of X.
(2) Let M be a closed n-dimensional manifold. Since M is compact Hausdorff it
is sufficient to show that no proper subspace of M is homeomorphic to M . Let
M1 , · · · , Mk be the connected components of M . Let N be a subspace of M that is
homeomorphic to M . Set Ni = N ∩ Mi . Since Ni is compact and Mi is Hausdorff,
Ni is a closed subset of Mi . We will show that Ni is an open subset of Mi by using
the invariance of domain theorem. Let y be a point in Ni . We will show that there is
an open set O of Mi with y ∈ O ⊂ Ni . Let U be an open neighbourhood of y in Mi
that is homeomorphic to an open set A of Rn . Let V be an open neighbourhood of y
in Ni that is homeomorphic to an open set B of Rn . Let f : A → U and g : B → V
be homeomorphisms. Since U ∩ V is an open subset of V , g −1 (U ∩ V ) is an open
set of Rn . By the invariance of domain theorem we see that f −1 (U ∩ V ) is an open
set of Rn . Therefore f −1 (U ∩ V ) is an open set of A. Since f is a homeomorphism
U ∩ V is an open set of U . Therefore U ∩ V is an open set of Mi . Thus O = U ∩ V
is the desired set. Since Mi is connected we have Ni = ∅ or Ni = Mi . Since N
has the same number of connected components as M we have Ni = Mi for every i.
Therefore N = M .
(3) Since G is compact Hausdorff, it is sufficient to show that no proper subspace
of G is homeomorphic to G. Let H be a subspace of G homeomorphic to G. For
each 1-simplex e of G, e ∩ H is a disjoint union of finitely many points and closed
intervals by Lemma 2.3. If there is a 1-simplex e of G with e ∩ H 6= e, then by
the assumption of G we have β1 (H) < β1 (G) or β0 (G \ int(e)) = β0 (G) + 1 where
β0 (X) is the number of connected component of X. Then it is easy to see that H
cannot be homeomorphic to G. Thus e ∩ H = e for each 1-simplex e of G. Since
β0 (H) = β0 (G), each 0-simplex of G is contained in H. Thus we have H = G as
desired. 
Let G be a one-dimensional simplicial complex. Let x be a point of G. The
degree of x in G, denoted by deg(x) = deg(x, G), is defined as follows. If x is a
0-simplex of G, then deg(x) is defined to be the number of 1-simplexes of G that
contain x. If x is an interior point of a 1-simplex of G, then deg(x) = 2. Note

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7

that deg(x, G′ ) = deg(x, G) for any subdivision G′ of G. Let br(G) be the set of all
points x of G with deg(x) = deg(x, G) ≥ 3.
Lemma 2.5. Let f : X → Y be a homeomorphism from a one-dimensional simplicial complex X to another one-dimensional simplicial complex Y . Let x be a point
of X. Then deg(x, X) = deg(f (x), Y ). Therefore f (br(X)) = br(Y ).
Proof. Suppose that deg(x, X) < deg(f (x), Y ). Let M be a sufficiently small
closed connected neighbourhood of f (x) in Y . Then the number of connected
components of M \ {f (x)} is deg(f (x), Y ). Let N be a sufficiently small closed
connected neighbourhood of x in X such that f (N ) ⊂ M . Then the number of
connected components of N \{x} is deg(x, X). Since f is a homeomorphism f (N ) is
also a neighbourhood of f (x) in Y . Each connected component of N \{x} is mapped
into a connected component of M \ {f (x)} under f . Therefore there is a component
of M \ {f (x)} that is disjoint from f (N ). Then f (N ) cannot be a neighbourhood
of f (x) in Y . This is a contradiction. Thus we have deg(x, X) ≥ deg(f (x), Y ).
Similarly, by considering f −1 we have deg(x, X) ≤ deg(f (x), Y ). 
Proposition 2.6. Let S be a set of finite one-dimensional simplicial complexes.
Suppose that each element of S has no isolated points and no connected components
each of which is homeomorphic to a closed interval. Then S is classed.
Proof. Let G1 and G2 be elements of S. Suppose that there are continuous
injections f1 : G1 → G2 and f2 : G2 → G1 . Let Hi be the maximal sub-complex of
Gi such that each 0-simplex of Hi is a face of at least two 1-simplexes of Hi for i =
1, 2. Namely Hi is the union of all subspaces of Gi each of which is homeomorphic
to a circle. If β1 (Gi ) = 0, then Hi is an empty set. Then β1 (Hi ) = β1 (Gi ) for
i = 1, 2. We may suppose without loss of generality that deg(x, Gi ) 6= 2 for each
0-simplex x contained in Gi \ Hi . Since G1 and G2 are compact Hausdorff, f1
and f2 are embeddings and Gi and fi (Gi ) are homeomorphic for i = 1, 2. By a
standard argument of combinatorial topology we have β1 (f1 (G1 )) = β1 (f1 (G1 ) ∩
H2 ). Suppose that H2 is not contained in f1 (G1 ). Then f1 (G1 ) ∩ H2 is a proper
subspace of H2 . Then we have β1 (f1 (G1 )∩H2 ) < β1 (H2 ) by a standard argument of
combinatorial topology. Thus β1 (G1 ) = β1 (f1 (G1 )) = β1 (f1 (G1 ) ∩ H2 ) is less than
β1 (H2 ) = β1 (G2 ). On the other hand β1 (G2 ) = β1 (f2 (G2 )) ≤ β1 (G1 ) by Lemma
2.3. This is a contradiction. Therefore H2 is contained in f1 (G1 ). Since f1 (G1 )
is a sub-complex of a subdivision of G2 , br(f1 (G1 )) is a subset of br(G2 ). Since
br(f1 (G1 )) = f1 (br(G1 )) we have |br(G1 )| ≤ |br(G2 )|. Similarly we have |br(G2 )| ≤
|br(G1 )|. Therefore br(f1 (G1 )) = br(G2 ). Similarly we have br(f2 (G2 )) = br(G1 ).
Let e be a 1-simplex of G2 that is not contained in H2 . Let x and y be the 0simplexes of G2 contained in e. Suppose that both deg(x, G2 ) and deg(y, G2 ) are
greater than 2. We will show that e is contained in f1 (G1 ). Suppose that e is not
contained in f1 (G1 ). By the condition br(f1 (G1 )) = br(G2 ) there is an interior point
z of e that is not contained in f1 (G1 ). Note that β0 (G2 \ {z}) = β0 (G2 ) + 1. Each
connected component of G2 \ {z} contains at least one point in br(G2 ). Therefore
β0 (f1 (G1 )) ≥ β0 (G2 \ {z}). Thus we have β0 (G1 ) = β0 (f1 (G1 )) is greater than
β0 (G2 ). Since each connected component of G1 contains at least one point in
br(G1 ) and br(f2 (G2 )) = br(G1 ), we have β0 (G2 ) = β0 (f2 (G2 )) ≥ β0 (G1 ). This is
a contradiction. Thus we have e ⊂ f1 (G1 ). Let e be a 1-simplex of G2 such that

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KOUKI TANIYAMA

one of the 0-simplexes of G2 contained in e, say x, has degree 1 in G2 . Note that by
the assumption on G2 the other 0-simplex of G2 contained in e, say y, has degree
greater than 2. Therefore y ∈ br(G2 ). Therefore e ∩ f1 (G1 ) is a neighbourhood of
y in e. By Lemma 2.3 and by the assumption on G1 , e ∩ f1 (G1 ) is homeomorphic
to a closed interval. Therefore f1 (G1 ) and G2 are homeomorphic. 
Proposition 2.7. Let n be a non-negative integer. Let X and Y be n-dimensional
finite simplicial complexes. Then X has finite multiplicity property over Y with
respect to map-multiplicity.
Proof. It is sufficient to show that there is a continuous map from f : X → Y
such that f −1 (y) is a finite set for every y ∈ Y . Let v1 , · · · , vk be the 0-simplexes
of X. Let s be an n-simplex of Y . Let x1 , · · · , xk be points in s that are in general
position. Namely any l points of them with l ≤ n + 1 are not contained in any
l − 2-dimensional affine subspace. Let f : X → Y be a map such that f (vi ) = xi
for each i and the restriction map of f to each simplex of X is an affine map. Then
f has the desired property. 
The following is an example of map-multiplicity of a topological space over another topological space. See [3] and [5] for other examples.
Proposition 2.8. Let n be a natural number and K2n+1 the 1-skeleton of a 2nsimplex. Let S1 be the unit circle. Then the following holds.
n(n + 1)
.
mmap (K2n+1 : S1 ) =
2
Proof. Let f : K2n+1 → S1 be the map constructed as follows. The image
of the 0-simplexes of K2n+1 under f forms the set of the vertices of a regular
(2n + 1)-gon inscribing to S1 . For each 1-simplex e of K2n+1 , f maps e homeomorphically onto an arc in S1 that is shorter than another arc in S1 with the same
n(n + 1)
end points. Then it is easy to check that mmap (f ) =
. Thus we have
2
n(n
+
1)
n(n
+
1)
. We will show mmap (g) ≥
for any continmmap (K2n+1 : S1 ) ≤
2
2
1
uous map g : K2n+1 → S . If g maps a 1-simplex to a point, then mmap (g) = ∞.
Thus we may suppose that g maps no 1-simplex to a point. We will deform g step
by step without increasing mmap (g) as follows. Let e be a 1-simplex of K2n+1 and
u and v the 0-simplexes contained in e. First suppose g(u) 6= g(v). Since g(e) is
a connected subset of S1 , g(e) contains an arc, say α, of S1 joining g(u) and g(v).
Then we modify g on e, still denoted by g, such that g maps e homeomorphically
onto α. Next suppose that g(u) = g(v). Since g does not map e to a point, there
is an interior point, say x, of e such that g(u) 6= g(x). Let e1 be the line segment
in e with end points u and x, and e2 the line segment in e with end points x and
v. Since g(ei ) is a connected subset of S1 , g(ei ) contains an arc, say βi , of S1
joining g(u) = g(v) and g(x) for i = 1, 2. Then we modify g on e such that g
maps ei homeomorphically onto βi for i = 1, 2. We perform this modification for
′
each 1-simplex of K2n+1 . Thus there is a subdivision, say K2n+1
of K2n+1 such
′
that g maps each 1-simplex of K2n+1 homeomorphically onto an arc in S1 . Then
′
by a slight modification near the 0-simplexes of K2n+1
we can modify g without
′
increasing mmap (g) such that no two 0-simplexes of K2n+1
are mapped to the same

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

9

point under g. Then we can deform g such that g maps each 1-simplex of K2n+1
homeomorphically onto an arc in S1 . Thus g maps the 0-simplexes of K2n+1 to
2n + 1 points in S1 and each 1-simplex of K2n+1 homeomorphically onto an arc
in S1 . Let V be the set of 0-simplexes of K2n+1 . Let e be a 1-simplex of K2n+1 .
Let d(e, g) = |g(e) ∩ g(V )| − 1. Let v be a 0-simplex of K2n+1 . Then the sum
of d(e, g) over all 1-simplexes e of K2n+1 containing v is greater than or equal to
2(1 + 2 + · · · + n). Therefore the sum S of d(e, g) over all 1-simplexes e of K2n+1
n(n + 1)(2n + 1)
2(1 + 2 + · · · + n)(2n + 1)
=
. Let
is greater than or equal to
2
2
1
1
y1 , · · · , y2n+1 be points in S \g(V ) such that each connected component of S \g(V )
contains exactly one of y1 , · · · , y2n+1 . Then d(e, g) = |g(e) ∩ {y1 , · · · , y2n+1 }|. Note
that |g −1 (yi )| is equal to the number of 1-simplexes e of K2n+1 with g(e) ∋ yi .
Therefore the sum |g −1 (y1 )| + · · · + |g −1 (y2n+1 )| is equal to S. Then by the pigeonn(n + 1)
n(n + 1)(2n + 1)
=
.
hole principle there is a point yi such that |g −1 (yi )| ≥
2(2n + 1)
2
n(n + 1)
. 
Thus we have mmap (g) ≥
2
3. Multiplicity of groups
Let G and H be groups and f : G → H a homomorphism from G to H. Then
the map-multiplicity of f is defined by mmap (f ) = sup{|f −1 (y)| | y ∈ H}. Note
that mmap (f ) = |Ker(f )| where |Ker(f )| is the order of the kernel Ker(f ) of f .
Therefore we denote mmap (f ) by mKer (f ) and call it kernel-multiplicity of f in this
section. Then the kernel-multiplicity mKer (G : H) of G over H is defined to be the
infimum of mKer (f ) where f varies over all homomorphisms from G to H.
We say that groups G and H are weakly isomorphic if there exists an injective
homomorphism from G to H and there exists an injective homomorphism from
H to G. We say that a set of groups G is classed if any two weakly isomorphic
groups in G are isomorphic. A group G is said to be co-Hopfian if every injective
homomorphism from G to G is an isomorphism.
Proposition 3.1. A set of co-Hopfian groups is classed.
Proof. Suppose that G and H are mutually weakly isomorphic co-Hopfian groups.
Let f : G → H and g : H → G be injective homomorphisms. Then g ◦ f : G → G is
an injective homomorphism. Since G is co-Hopfian, g ◦ f is an isomorphism. Hence
g is surjective and therefore g is an isomorphism from H to G. 
Theorem 3.2. Let G be a classed set of groups. Let CG be a category whose objects
are the elements of G and morphisms are the homomorphisms between the elements
of G. Let mKer be the kernel-multiplicity on CG and dmKer the kernel-multiplicity
distance. Let G and H be elements of G. Then G and H are isomorphic if and
only if dmKer (G, H) = 0. Thus, if mKer has finite multiplicity property on G, then
the pseudo distance dmKer defines a distance on the set of isomorphism classes of
G.
The proof of Theorem 3.2 is entirely analogous to that of Theorem 2.2 and we
omit it. Note that a finite group is co-Hopfian, and kernel-multiplicity has finite

10

KOUKI TANIYAMA

multiplicity property on the set of all finite groups. Thus dmKer defines a distance
on the set of isomorphism classes of all finite groups.
We now define another multiplicity for group homomorphisms. Let G and H be
groups and f : G → H a homomorphism from G to H. Then the cokernel of f
is the quotient set Coker(f ) = H/f (G). Then we call mCoker (f ) = |Coker(f )| the
cokernel-multiplicity of f .
Proposition 3.3. A cokernel-multiplicity mCoker is a multiplicity. Namely the
following (1) and (2) hold.
(1) mCoker (idG ) = 1 for any group G.
(2) For any group homomorphisms f : G → H and g : H → K, mCoker (g ◦ f ) ≤
mCoker (f )mCoker (g).
Proof. (1) Since Coker(idG ) = G/G is a trivial group, mCoker (idG ) = |G/G| = 1.
(2) Since Coker(g ◦ f ) = K/g ◦ f (G) = K/g(f (G)), mCoker (g ◦ f ) = |Coker(g ◦ f )| =
|K/g(f (G))| = |K/g(H)| · |g(H)/g(f (G))|. Since |K/g(H)| = mCoker (g) it is sufficient to show |g(H)/g(f (G))| ≤ mCoker (f ) = |H/f (G)|. There is a surjection from
H/f (G) to g(H)/g(f (G)) induced by g. Therefore |g(H)/g(f (G))| ≤ |H/f (G)|. 
The following arguments show an analogy of cokernel-multiplicity to kernelmultiplicity. The proofs are entirely analogous and we omit them. We say that
groups G and H are co-weakly isomorphic if there exists a surjective homomorphism from G to H and there exists a surjective homomorphism from H to G. We
say that a set of groups G is co-classed if any two co-weakly isomorphic groups in G
are isomorphic. A group G is said to be Hopfian if every surjective homomorphism
from G to G is an isomorphism.
Proposition 3.4. A set of Hopfian groups is co-classed.
Theorem 3.5. Let G be a co-classed set of groups. Let CG be a category whose objects are the elements of G and morphisms are the homomorphisms between the elements of G. Let mCoker be the cokernel-multiplicity on CG and dmCoker the cokernelmultiplicity distance. Let G and H be elements of G. Then G and H are isomorphic
if and only if dmCoker (G, H) = 0. Thus, if mCoker has finite multiplicity property on
G, then the pseudo distance dmCoker defines a distance on the set of isomorphism
classes of G.
The following example shows a difference between kernel-multiplicity and cokernelmultiplicity. Let F (n) be a free group of rank n. Then there exists an injective
homomorphism from F (2) to F (3), and a surjective homomorphism from F (3) to
F (2). Moreover it is well-known that there exist an injective homomorphism from
F (3) to F (2). Suppose that f : F (2) → F (3) is a homomorphism. Let F A(n) be a
free abelian group of rank n and αn : F (n) → F A(n) the abelianization homomorphism. Then there is a homomorphism fˆ : F A(2) → F A(3) such that α3 ◦ f = fˆ ◦
α2 . Then |F (3)/f (F (2))| ≥ |α3 (F (3))/α3 (f (F (2)))| = |α3 (F (3))/(fˆ(α2 (F (2)))| =
|F A(3)/fˆ(F A(2))| = ∞. Therefore mCoker (f ) = |F (3)/f (F (2))| = ∞. After all
we have mKer (F (2) : F (3)) = mKer (F (3) : F (2)) = 1, mCoker (F (2) : F (3)) = ∞
and mCoker (F (3) : F (2)) = 1. Therefore we have dmKer (F (2), F (3)) = 0 and
dmCoker (F (2), F (3)) = ∞.

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

11

However we have the following proposition.
Proposition 3.6. Let G and H be finite groups. Then dmKer (G, H) = dmCoker (G, H).
Proof. Since dmKer (G, H) = loge (mKer (G : H)mKer (H : G)) and dmCoker (G, H) =
loge (mCoker (G : H)mCoker (H : G)) it is sufficient to show mKer (G : H)mKer (H :
G) = mCoker (G : H)mCoker (H : G). Let f : G → H be a homomorphism. Since
G/Ker(f ) is isomorphic to f (G) we have |G|/|Ker(f )| = |f (G)|. On the other hand
|Coker(f )| = |H|/|f (G)|. Thus we have
|Ker(f )| =

|G|
|Coker(f )|.
|H|

Therefore |Ker(f )| = mKer (G : H) if and only if |Coker(f )| = mCoker (G : H).
Similarly, for a homomorphism g : G → H we have
|Ker(g)| =

|H|
|Coker(g)|
|G|

and |Ker(g)| = mKer (H : G) if and only if |Coker(g)| = mCoker (H : G). Thus
taking such f and g we have
mKer (G : H)mKer (H : G) = |Ker(f )| · |Ker(g)| =

|G|
|H|
|Coker(f )|
|Coker(g)|
|H|
|G|

= |Coker(f )| · |Coker(g)| = mCoker (G : H)mCoker (H : G).
This completes the proof. 
4. Multiplicity of finitely generated modules over a principal ideal
domain
Throughout this section R is a principal ideal domain. Some of the results in
this section hold for any unitary ring R. See Remark 4.14. However we restrict our
attention to a principal ideal domain for the simplicity. Let M be an R-module
finitely generated over R. We denote the minimal number of generators of M over
R by r(M ). For M = {0} we define r(M ) = 0. Note that when R is a field, M is a
finite dimensional vector space over R whose dimension dim(M ) = r(M ). Let N be
an R-module finitely generated over R and f : M → N an R-linear map from M to
N . By Ker(f ) we denote the kernel of f and by Coker(f ) = N/f (M ) we denote the
cokernel of f . We define the kernel-rank-multiplicity of f by mr(Ker) (f ) = er(Ker(f ))
and the cokernel-rank-multiplicity of f by mr(Coker) (f ) = er(Coker(f )) .
Proposition 4.1. (1) A kernel-rank-multiplicity mr(Ker) is a multiplicity.
(2) A cokernel-rank-multiplicity mr(Coker) is a multiplicity.
For the proof of Proposition 4.1 we prepare the following proposition.
Proposition 4.2. Let M , N and L be R-modules finitely generated over R. Let
f : M → N and g : N → L be R-linear maps. Then the following (1) and (2) hold.
(1) r(Ker(g ◦ f )) ≤ r(Ker(f )) + r(Ker(g)).
(2) r(Coker(g ◦ f )) ≤ r(Coker(f )) + r(Coker(g)).

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KOUKI TANIYAMA

For the proof of Proposition 4.2, we need the following lemmas. Since they are
standard facts in module theory, we omit the proofs. See for example [1, Chapter
3] for the proofs.
Lemma 4.3. Let M and N be R-modules. Suppose that M is finitely generated
over R and there exist a surjective R-linear map from M to N . Then N is finitely
generated over R and r(N ) ≤ r(M ).
Lemma 4.4. Let M be an R-module finitely generated over R and N an Rsubmodule of M . Then N is finitely generated over R and r(N ) ≤ r(M ).
Lemma 4.5. Let M and N be R-modules and f : M → N a surjective R-linear
map. Suppose that the R-modules Ker(f ) and N are finitely generated over R.
Then M is finitely generated over R and r(M ) ≤ r(Ker(f )) + r(N ).
Proof of Proposition 4.2. (1) Since Ker(g ◦ f ) = f −1 (Ker(g)) = f −1 (Ker(g) ∩
f (M )), f |Ker(g◦f ) : Ker(g ◦ f ) → Ker(g) ∩ f (M ) is a surjective R-linear map whose
kernel Ker(f |Ker(g◦f ) ) is equal to Ker(f ). By Lemma 4.4 r(Ker(g) ∩ f (M )) ≤
r(Ker(g)). By Lemma 4.5 r(Ker(g ◦ f )) ≤ r(Ker(f |Ker(g◦f ) )) + r(Ker(g) ∩ f (M )).
Thus we have r(Ker(g◦f )) ≤ r(Ker(f ))+r(Ker(g)∩f (M )) ≤ r(Ker(f ))+r(Ker(g)).
(2) Note that Coker(g ◦ f ) = L/g(f (M )) and L ⊃ g(N ) ⊃ g(f (M )). By the third
isomorphism theorem
(L/g(f (M )))/(g(N )/g(f (M ))) ∼
= L/g(N ).
Then by Lemma 4.5 r(Coker(g ◦ f )) = r(L/g(f (M ))) ≤ r(g(N )/g(f (M ))) +
r(L/g(N )) = r(g(N )/g(f (M ))) + r(Coker(g)). There exists a surjective R-linear
map from Coker(f ) = N/f (M ) to g(N )/g(f (M )) induced by g. Therefore we
have r(g(N )/g(f (M ))) ≤ r(Coker(f )) by Lemma 4.3. Thus we have the desired
inequality. 
Proof of Proposition 4.1. Let M , N and L be R-modules finitely generated over
R and let f : M → N and g : N → L be R-linear maps.
(1) Since r(Ker(idM )) = r({0}) = 0, we have mr(Ker) (idM ) = e0 = 1. By Proposition 4.2 (1) we have r(Ker(g ◦ f )) ≤ r(Ker(f )) + r(Ker(g)). Therefore we have
mr(Ker) (g ◦ f ) = er(Ker(g◦f )) ≤ er(Ker(f )) er(Ker(g)) = mr(Ker) (f )mr(Ker) (g).

(2) Since r(Coker(idM )) = r({0}) = 0, we have mr(Coker) (idM ) = e0 = 1. By
Proposition 4.2 (2) we have r(Coker(g ◦ f )) ≤ r(Coker(f ))+ r(Coker(g)). Therefore
we have
mr(Coker) (g◦f ) = er(Coker(g◦f )) ≤ er(Coker(f )) er(Coker(g)) = mr(Coker) (f )mr(Coker) (g).
This completes the proof. 
By Lemma 4.4 and Lemma 4.3 we have both mr(Ker) and mr(Coker) have finite
multiplicity property in the category of R-modules finitely generated over R and
R-linear maps. Thus we have two pseudo distances dmr(Ker) and dmr(Coker) . The
following example shows a difference between them. Let R = Z be the ring of
integers and nZ the ideal of Z generated by an integer n. Then both Z and Z/2Z
are Z-modules finitely generated over Z. It is easy to see mr(Ker) (Z : Z/2Z) = e,

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

13

mr(Ker) (Z/2Z : Z) = e, mr(Coker) (Z : Z/2Z) = 1 and mr(Coker) (Z/2Z : Z) = e.
Therefore we have dmr(Ker) (Z, Z/2Z) = 2 and dmr(Coker) (Z, Z/2Z) = 1.
In contrast to the example above, we have the following result for free R-modules
finitely generated over R. Let n be a non-negative integer. By Rn = R ⊕ · · · ⊕ R,
|
{z
}
n

we denote a free R-module of rank r(Rn ) = n. Here R0 = {0} denotes a zeromodule.
Proposition 4.6. Let n and m be non-negative integers. Then
dmr(Ker) (Rn , Rm ) = dmr(Coker) (Rn , Rm ) = |m − n|.
Proof. We may suppose without loss of generality that m ≤ n. Then it is easy to
see mr(Ker) (Rm : Rn ) = 1, mr(Ker) (Rn : Rm ) ≤ en−m , mr(Coker) (Rm : Rn ) ≤ en−m
and mr(Coker) (Rn : Rm ) = 1. Let f : Rn → Rm be an R-linear map. Then by
Lemma 4.5 we have r(Rn ) ≤ r(Ker(f )) + r(f (Rn )). Therefore r(Ker(f )) ≥ r(Rn ) −
r(f (Rn )). Since r(f (Rn )) ≤ r(Rm ) by Lemma 4.4 we have r(Ker(f )) ≥ r(Rn ) −
r((Rm ) = n − m. Therefore mr(Ker) (Rn : Rm ) ≥ en−m . Hence mr(Ker) (Rn : Rm ) =
en−m . Let g : Rm → Rn be an R-linear map. Then by Lemma 4.3 r(g(Rm )) ≤
r(Rm ) = m. By Lemma 4.5 we have n = r(Rn ) ≤ r(g(Rm )) + r(Rn /g(Rm )).
Therefore r(Rn /g(Rm )) ≥ n − m. Therefore mr(Coker) (Rm : Rn ) ≥ en−m . Hence
mr(Coker) (Rm : Rn ) = en−m . Then by a calculation we have the result. 
The following is an immediate corollary of Proposition 4.6.
Corollary 4.7. Let R be a field and V and W finite dimensional vector spaces
over R. Then
dmr(Ker) (V, W ) = dmr(Coker) (V, W ) = |dim(V ) − dim(W )|.
Theorem 4.8. Let R be a principal ideal domain. Let OR be a set of R-modules
finitely generated over R. Let CR be a category whose objects are the elements of
OR and whose morphisms are R-linear maps between the elements of OR .
(1) Let mr(Ker) be the kernel-rank-multiplicity on CR and dmr(Ker) the kernel-rankmultiplicity distance. Let M and N be elements of OR . Then M and N are
isomorphic if and only if dmr(Ker) (M, N ) = 0. Thus the pseudo distance dmr(Ker)
defines a distance on the set of isomorphism classes of OR .
(2) Let mr(Coker) be the cokernel-rank-multiplicity on CR and dmr(Coker) the cokernelrank-multiplicity distance. Let M and N be elements of OR . Then M and N are
isomorphic if and only if dmr(Coker) (M, N ) = 0. Thus the pseudo distance dmr(Coker)
defines a distance on the set of isomorphism classes of OR .
The essential part of Theorem 4.8 is the following proposition.
Proposition 4.9. Let M and N be R-modules finitely generated over R.
(1) Suppose that there exist injective R-linear maps f : M → N and g : N → M .
Then M and N are isomorphic.
(2) Suppose that there exist surjective R-linear maps f : M → N and g : N → M .
Then both f and g are injective, and M and N are isomorphic.

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KOUKI TANIYAMA

For the proof of Proposition 4.9, we prepare some lemmas. For an R-module M ,
we denote the torsion submodule of M by tor(M ).
Lemma 4.10. Let M and N be R-modules and f : M → N an R-linear map.
Then f (tor(M )) ⊂ tor(N ).
Proof. Suppose x ∈ tor(M ). Then there is a non-zero element r ∈ R such that
rx = 0. Then rf (x) = f (rx) = f (0) = 0. Therefore f (x) ∈ tor(N ). 
Thus for f : M → N , an induced R-linear map fˆ : M/tor(M ) → N/tor(N ) is
defined by fˆ([x]) = [f (x)] where x ∈ M and [x] ∈ M/tor(M ) denotes the set of
elements y ∈ M with x − y ∈ tor(M ).
Lemma 4.11. Let M and N be R-modules and f : M → N an injective R-linear
map. Let fˆ : M/tor(M ) → N/tor(N ) be an induced R-linear map. Then fˆ is
injective.
Proof. Suppose fˆ([x]) = fˆ([y]). Then [f (x)] = [f (y)] and therefore f (x) − f (y) =
f (x− y) ∈ tor(N ). Let r ∈ R such that rf (x− y) = 0. Since rf (x− y) = f (r(x− y))
and f is injective we have r(x − y) = 0. Thus x − y ∈ tor(M ) and [x] = [y]. 
For an element r of R, we denote the ideal generated by r by (r). Then the
quotient R/(r) is a torsion R-module unless r = 0. The following Lemma 4.12 and
Proposition 4.13 are known facts. However we give a direct elementary proof of
Lemma 4.12 for convenience.
Lemma 4.12. Let p be a prime element of R and k a natural number. Then an
R-module R/(pk ) is Artinian.
Proof. It is sufficient to show that a proper R-submodule M of R/(pk ) is isomorphic to R/(pj ) for some non-negative integer j < k. Let ϕ : R → R/(pk ) be a
canonical projection. Then ϕ−1 (M ) is an R-submodule, hence an ideal of R. Let
a be an element of R such that ϕ−1 (M ) = (a). Since (a) ⊃ (pk ) we have (a) = (pl )
for some non-negative integer l ≤ k. If l = 0 then ϕ−1 (M ) = (1R ) = R where 1R
is the multiplicative identity of R and then M = ϕ(R) = R/(pk ) is not a proper
R-submodule. Thus we have l ≥ 1. Let ψ : R → R be an R-linear map defined
by ψ(x) = pl x. Then ψ induces an R-linear map ψ̂ : R/(pk−l ) → R/(pk ). By the
fact that R is a unique factorization domain it follows that ψ is injective. By the
definition of ψ̂ we have ψ̂(R/(pk−l )) = M . Therefore M is isomorphic to R/(pk−l ).

It is well-known that a direct sum of finitely many Artinian R-modules is Artinian. By the structure theorem, a torsion R-module finitely generated over R is
a direct sum of finitely many cyclic R-modules of the form R/(pk ). Therefore we
have the following proposition.
Proposition 4.13. A torsion R-module finitely generated over R is Artinian.
Proof of Proposition 4.9. (1) By Lemma 4.11 the free R-modules M/tor(M )
and N/tor(N ) have the same rank. Therefore they are isomorphic. By Lemma

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

15

4.10 we have injective R-linear maps f ′ : tor(M ) → tor(N ) and g ′ : tor(N ) →
tor(M ). Then g ′ ◦ f ′ : tor(M ) → tor(M ) is an injective R-linear map. Since
tor(M ) is Artinian, it is co-Hopfian. Therefore g ′ ◦ f ′ is surjective. Thus g ′ is a
surjection, hence an isomorphism. Then by the structure theorem of R-modules
finitely generated over R, M and N are isomorphic.
(2) Note that g ◦ f : M → M is a surjective R-linear map. It is well-known that
M is Noetherian, hence Hopfian. Therefore g ◦ f is injective. Therefore f is an
injection, hence an isomorphism. Similarly, g is also an isomorphism. 
Proof of Theorem 4.8. (1) The ‘only if’ part is clear. Suppose dmr(Ker) (M, N ) =
0. Then there are injective R-linear maps f : M → N and g : N → M . Then by
Proposition 4.9 (1) M and N are isomorphic.
(2) The ‘only if’ part is clear. Suppose dmr(Coker) (M, N ) = 0. Then there are
surjective R-linear maps f : M → N and g : N → M . Then by Proposition 4.9 (2)
M and N are isomorphic. 
Remark 4.14. The main reason that we have restricted our attention to the case
R is a principal ideal domain is the fact that Lemma 4.4 does not hold in general
when R is not a principal ideal domain. Let R = Z[X] be a polynomial ring over
the integer ring Z. Let M = R and N = (2, X) where (2, X) denotes the ideal of
R generated by 2 and X. Then r(M ) = 1. Since N is an ideal of an R-module M ,
N is an R-submodule of M . Since N is not a principal ideal we have r(N ) = 2.
Actually Proposition 4.2 (1) does not hold in general when R is not a principal ideal
domain. Let L = {0} be the zero module. Let f : N → M be the inclusion map and
g : M → L the zero map. Then r(Ker(g ◦ f )) = r(N ) = 2, r(Ker(f )) = r({0}) = 0
and r(Ker(g)) = r(M ) = 1. Moreover Proposition 4.9 (1) does not hold in general
when R is not a principal ideal domain. Let f : M → N be an R-linear map defined
by f (x) = 2x and g : N → M the inclusion R-linear map. Then they are injective.
Since r(M ) 6= r(N ) they are not isomorphic.
We note however Lemma 4.3 and Lemma 4.5 hold for any unitary ring R. Therefore Proposition 4.2 (2) and Proposition 4.1 (2) also hold for any unitary ring R.
We also note that Proposition 4.9 (2) is true when M or N is a Hopfian Rmodule.
5. Multiplicity of knots
In this section we define a category of oriented knot types in 3-space and define a
multiplicity on it. Then we study its multiplicity distance. Throughout this section
we work in the piecewise linear category. First we define a category which we call
a neighbourhood category of oriented knots.
For a manifold X with boundary, we denote the interior of X by int(X) and
the boundary of X by ∂(X) respectively. By D2 we denote the unit disk in the
2-dimensional Euclidean space R2 and by S1 = ∂(D2 ) we denote the unit circle
in R2 . A core of a solid torus V is a circle c ⊂ int(V ) such that the pair (V, c)
is homeomorphic to the pair (D2 × S1 , (0, 0) × S1 ). Two oriented cores c1 and c2
of V are equivalent if they are ambient isotopic in V as oriented circles. A coreorientation of V is an equivalence class of oriented cores in V . A solid torus V is
core-oriented if a core-orientation of V is specified. Let V be a core-oriented solid
torus embedded in S3 and k an oriented circle contained in int(V ). Then the pair

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KOUKI TANIYAMA

(V, k) is called a nesting. Two nestings (V, k) and (V ′ , k ′ ) are equivalent if there
exist an orientation preserving homeomorphism ϕ : S3 → S3 such that ϕ(V ) = V ′
respecting core-orientations and ϕ(k) = k ′ respecting orientations. The equivalence
class containing (V, k) is denoted by [(V, k)]. Let K be the set of all oriented knot
types in the 3-sphere S3 . Let K1 and K2 be elements of K. Namely K1 and K2
are oriented knot types in S3 . A morphism f from K1 to K2 is an equivalence class
[(V, k)] of nestings such that the knot type of an oriented core of V is K2 and the
knot type of k is K1 . Then we denote f : K1 → K2 and f = [(V, k)].
Let K3 be an element of K and g = [(W, l)] a morphism from K2 to K3 . Let
N ⊂ int(W ) be a regular neighbourhood of l. Then N is a solid torus and l is a
core of N . We give a core-orientation of N by the orientation of l. Let f : S3 → S3
be an orientation preserving homeomorphism such that f (V ) = N respecting coreorientations. Then f (k) is an oriented knot contained in W whose knot type is K1 .
Thus [(W, f (k))] is a morphism from K1 to K3 . We define the composition of f
and g by g ◦ f = [(W, f (k))].
We will check the well-definedness of composition as follows. Suppose that an
orientation preserving homeomorphism ϕ : S3 → S3 maps (V, k) to (V ′ , k ′ ) and an
orientation preserving homeomorphism ψ : S3 → S3 maps (W, l) to (W ′ , l′ ). Let
N ′ ⊂ int(W ′ ) be a regular neighbourhood of l′ and let f ′ : S3 → S3 be an orientation
preserving homeomorphism such that f (V ′ ) = N ′ respecting core-orientations. It
is sufficient to show [(W, f (k))] = [(W ′ , f ′ (k ′ ))]. Since [(W, f (k))] = [W ′ , ψ ◦ f (k))]
by ψ, it is sufficient to show that ψ ◦ f (k) is ambient isotopic to f ′ (k ′ ) in W ′ .
Note that both ψ(N ) and N ′ are solid tori in int(W ′ ) with the same oriented core
l′ . Therefore there is an isotopy ht : W ′ → W ′ with 0 ≤ t ≤ 1 and h0 = idW ′
point-wisely fixed on ∂(W ′ ) ∪ l′ such that h1 (ψ(N )) = N ′ . Let γ : S3 → S3 be an
extension of h1 : W ′ → W ′ such that γ(x) = x for any x ∈ S3 \ W ′ . Then ψ ◦ f (k)
is ambient isotopic to h1 (ψ ◦ f (k)) = γ(ψ ◦ f (k)) = γ ◦ ψ ◦ f (k). Thus it is sufficient
to show that γ ◦ ψ ◦ f (k) is ambient isotopic to f ′ (k ′ ) in W ′ . We now have the
following sequence of orientation preserving self-homeomorphisms of S3 .
f′

ϕ

f

ψ

γ

(N ′ , f ′ (k ′ )) ← (V ′ , k ′ ) ← (V, k) → (N, f (k)) → (ψ(N ), ψ ◦f (k)) → (N ′ , γ ◦ψ ◦f (k))

Let η : N ′ → N ′ be the restriction of γ ◦ ψ ◦ f ◦ ϕ−1 ◦ (f ′ )−1 to N ′ . Then η is an
orientation preserving homeomorphism also preserving the core-orientation of N ′ .
Since η is a restriction of a self-homeomorphism of S3 , η maps a preferred longitude
of N ′ to a preferred longitude of N ′ . Therefore η is isotopic to idN ′ , see for example
[8]. Therefore f ′ (k ′ ) is ambient isotopic to η(f ′ (k ′ )) = γ ◦ ψ ◦ f (k) in N ′ , hence in
W ′ as desired.
An example of the composition of two morphisms is illustrated in Figure 5.1.
Let K be an element of K. Let k ⊂ S3 be an oriented knot whose knot type
is K and N a regular neighbourhood of k in S3 . We define idK : K → K by
idK = [(N, k)]. It is clear that idK actually satisfies the axiom of identity morphism
of category. By the definition other axioms of category clearly hold. Thus we have a
category whose objects are the elements of K. We call this category a neighbourhood
category of oriented knots and denote it by CK .

Remark 5.1. In [6] Ryo Nikkuni has pointed out the following. Suppose that f =
[(V, k)] is a morphism from K1 to K2 . Let N ⊂ int(V ) be a regular neighbourhood
of k. Let E2 = S3 \ int(V ) and E1 = S3 \ int(N ). Then Ei is an exterior manifold
of a knot representing Ki for i = 1, 2 and we have an inclusion E2 ⊂ E1 . Then the

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

17

Figure 5.1.
inclusion induces a homomorphism from π1 (E2 ) to π1 (E1 ) where π1 (X) denotes
the fundamental group of X.
For any elements K1 and K2 of K, there is a morphism c : K1 → K2 with
c = [(N, k)] where N is a core-oriented solid torus in S3 whose knot type is K2 and
k is an oriented knot contained in a 3-ball in N whose knot type is K1 . We call c
the null morphism from K1 to K2 . By definition it is clear that both c ◦ f and g ◦ c
are again null morphisms for any f and g. For the existence of non-null morphisms,
we have the following proposition. For an oriented knot k in S3 , −k denotes an
oriented knot obtained from k by reversing its orientation. For an element K of K
represented by an oriented knot k in S3 , −K denotes an element of K represented
by −k. A morphism f = [(V, k)] from K1 to K2 is called an inversion if k is a core
of V and the core-orientation of V is represented by −k. Thus if f : K1 → K2 is
an inversion then K2 = −K1 .
Proposition 5.2. Let K1 and K2 be elements of K. Suppose that there exists a
non-null morphism f : K1 → K2 . Then at least one of the following holds.
(1) K1 = K2 and f = idK1 .
(2) K1 = −K2 and f is an inversion.
(3) K2 is the trivial knot type.
(4) K1 is a satellite knot type with companion knot type K2 .
Proof. Suppose f = [(V, k)]. Suppose that K2 is non-trivial. Then V is a knotted
solid torus in S3 . Since k is not contained in any 3-ball in V , k is essential in V .
Then either k is a core of V , then we have (1) or (2), or k is a satellite knot with
companion solid torus V , then we have (4). 
Let ϕ : X → Y be a continuous map from a circle X to a 1-dimensional manifold
Y without boundary. A point x ∈ X is a regular point of ϕ if there is a neighbourhood U of x in X such that ϕ|U is injective. A point x ∈ X is a critical point
of ϕ if it is not a regular point. By cr(ϕ) we denote the set of all critical points

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KOUKI TANIYAMA

of ϕ. We say that ϕ is simple if X = I1 ∪ · · · ∪ In for some n ∈ N and closed
intervals I1 , · · · , In such that ϕ|Ii is injective for each i ∈ {1, · · · , n}. We say that
ϕ is generic if it is simple and ϕ|cr(ϕ) is injective.
We now define a multiplicity on the neighbourhood category of oriented knots as
follows. Let K1 and K2 be elements of K and and f = [(V, k)] a morphism from K1
to K2 . Let h : V → S1 × D2 be a homeomorphism and π : S1 × D2 → S1 a natural
projection. We say that h is generic with respect to k if π ◦ h|k : k → S1 is generic.
Let m(h, k) = max{|(π ◦ h|k )−1 (y)| | y ∈ S1 }. Note that if y ∈ π ◦ h(cr(π ◦ h|k ))
then |(π ◦ h|k )−1 (y)| < m(h, k). Then we define the multiplicity of f , denoted
by mK (f ), to be the minimum of m(h, k) where h : V → S1 × D2 varies over
all homeomorphisms that are generic with respect to k. By definition mK (f ) is
well-defined. We call mK (f ) the knot-multiplicity of f .
Proposition 5.3. The knot-multiplicity mK is a multiplicity on the neighbourhood
category of oriented knots.
Proof. It is clear that mK (idK ) = 1 for any element K of K. Let f : K1 → K2 and
g : K2 → K3 be morphisms where f = [(V, k)] and g = [(W, l)]. Let h : V → S1 ×D2
be a homeomorphism that is generic with respect to k with m(h, k) = mK (f )
and s : W → S1 × D2 a homeomorphism that is generic with respect to l with
m(s, l) = mK (g). Let N ⊂ int(W ) be a regular neighbourhood of l and f : S3 → S3
an orientation preserving homeomorphism such that f (V ) = N respecting coreorientations. Then the composition g ◦ f : K1 → K3 is given by g ◦ f = [(W, f (k))].
By a modification of s on N we may assume that f (cr(π ◦ h|k )) is a subset of
cr(π ◦ s|f (k) ) and away from a small neighbourhood, say U , of cr(π ◦ s|l ), and all
other critical points cr(π ◦ s|f (k) ) \ f (cr(π ◦ h|k )) are contained in U . See Figure
5.2 which illustrates a part of W containing a critical point, say z, of π ◦ s|l .
Let yz be a point in S1 as indicated in Figure 5.2. Then |(π ◦ s|f (k) )−1 (yz )| is
locally maximal in a small neighbourhood of yz in S1 . Therefore we only need
to consider |(π ◦ s|f (k) )−1 (yz )| for each critical point z of π ◦ s|l . Therefore we
have m(s, f (k)) ≤ m(h, k)m(s, l). Thus mK (g ◦ f ) ≤ m(s, f (k)) ≤ mK (f )mK (g) as
desired. 

Figure 5.2.

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

19

Proposition 5.4. Let K1 and K2 be elements of K.
(1) mK (K1 : K2 ) = 1 if and only if K1 = K2 or K1 = −K2 .
(2) mK (K1 : K2 ) = 2 if and only if K1 is trivial and K2 is non-trivial, or K1 is
non-trivial and K1 is a (2, p)-cable knot of K2 or −K2 for some odd number p.
(When K2 is trivial, K1 is a non-trivial (2, p)-torus knot.)
Proof. (1) Suppose mK (K1 : K2 ) = 1. Then there exists a morphism f = [(V, k)]
from K1 to K2 with mK (f ) = 1. Then k is a core of V and hence we have K1 = K2
or K1 = −K2 . The converse is similar.
(2) Suppose mK (K1 : K2 ) = 2. Then there exists a morphism f = [(V, k)] from K1
to K2 with mK (f ) = 2. Then the number of critical points of k with respect to the
projection from V to S1 is 0 or 2. In the first case k is a (2, p)-cabling of the core of
V . Therefore K1 is a (2, p)-cable knot of K2 or −K2 for some odd number p. If K2
is non-trivial, then K1 is also non-trivial. If K2 is trivial, mK (K1 : K2 ) 6= 1 implies
K1 is non-trivial. In the second case k is a trivial knot. Then mK (K1 : K2 ) 6= 1
implies K2 is non-trivial. The converse is clear. 
Theorem 5.5. Let CK be a neighbourhood category of oriented knots. Let mK be
the knot-multiplicity on CK and dmK the knot-multiplicity distance. Let K1 and K2
be elements of K. Then dmK (K1 , K2 ) = 0 if and only if K1 = K2 or K1 = −K2 .
Thus the pseudo distance dmK defines a distance on the set of unoriented knot types
in S3 .
Proof. Note that dmK (K1 , K2 ) = 0 if and only if mK (K1 : K2 ) = mK (K2 : K1 ) =
1. Then by Proposition 5.4 (1) we have the conclusion. 
We now discuss several relations between knot-multiplicity and some geometric knot invariants as follows. In [7] Ozawa defined a trunk of a knot as follows. Let k be an oriented knot in S3 \ {(0, 0, 0, 1), (0, 0, 0, −1)}. Let ϕ : S3 \
{(0, 0, 0, 1), (0, 0, 0, −1)} → S2 × R be a fixed homeomorphism and π : S2 × R → R
a natural projection. We say that k is in general position if π ◦ ϕ|k : k → R
is generic. Then the trunk of k, denoted by trunk(k), is defined by trunk(k) =
max{|(π ◦ ϕ|k )−1 (y)||y ∈ R}. The trunk of an oriented knot type K, denoted by
trunk(K), is defined to be the minimum of trunk(k) where k varies over all oriented
knots in S3 in general position whose oriented knot type is K.
Proposition 5.6. Let K1 and K2 be elements of K. Then

trunk(K1 )
≤ mK (K1 : K2 ) ≤ trunk(K1 ).
trunk(K2 )

If K2 is non-trivial, K2 6= K1 , K2 6= −K1 and not a companion of K1 , then
mK (K1 : K2 ) = trunk(K1 ).
Proof. Let k2 be an oriented knot in S3 representing K2 with trunk(k2 ) =
trunk(K2 ). Let N be a regular neighbourhood of k2 and f = [(N, k1 )] a morphism
from K1 to K2 with mK (f ) = mK (K1 : K2 ). Then by deforming k1 if necessary, we
have trunk(k1 ) ≤ trunk(k2 )mK (f ). Thus we have trunk(K1 ) ≤ trunk(K2 )mK (K1 :
K2 ). Therefore the first inequality holds. Let c = [(V, k)] be a null morphism from
K1 to K2 . Clearly mK (c) ≤ trunk(K1 ). Therefore the second inequality holds.

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KOUKI TANIYAMA

Conversely, since k is contained in a 3-ball in V , k lifts to the universal covering
space Ṽ of V . Then the projection from V to S1 lifts to the projection from Ṽ to
R. Therefore we have trunk(K1 ) ≤ mK (c). Then the final statement follows by
Proposition 5.2. 
As we have observed, there are not so many morphisms when the target knot
is non-trivial. Therefore, for an element K of K, we define m(K) = mK (K : T )
where T is the trivial knot type, and call it the multiplicity index of K. Since
trunk(T ) = 2, we have the following corollary of Proposition 5.6.
Corollary 5.7. Let K be an element of K. Then m(K) ≥

trunk(K)
.
2

If K 6= T then mK (T : K) = trunk(T ) = 2. Therefore
trunk(K)
· 2 = loge trunk(K).
2
In [7] it is shown that for any natural number n, there exist an oriented knot type
K with trunk(K) ≥ n. Therefore we have the following proposition.
dmK (K, T ) ≥ loge

Proposition 5.8. The pseudo metric space (K, dmK ) is unbounded.
Let braid(K) be the braid index of K and bridge(K) the bridge index of K.
Proposition 5.9. Let K be an element of K.
(1) m(K) ≤ braid(K).
(2) m(K) ≤ 2 · bridge(K) − 1.
Proof. (1) Suppose braid(K) = n. Then K has a representative k contained in
an unknotted solid torus V ⊂ S3 such that k intersects each meridian disk of V at
exactly n points. Then f = [(V, k)] is a morphism from K to T with mK (f ) = n.
(2) Let B1 and B2 be 3-balls such that S3 = B1 ∪ B2 and B1 ∩ B2 = ∂(B1 ) =
∂(B2 ). Suppose bridge(K) = n. Then there is a representative k of K such that
both (B1 , B1 ∩ k) and (B2 , B2 ∩ k) are trivial n-string tangles. Choose a point
x ∈ ∂(B1 ) ∩ k. Let γ be a circle on ∂(B1 ) \ k separating x from other 2n − 1
points of ∂(B1 ) ∩ k. Let N be a regular neighbourhood of γ in S3 with N ∩ k = ∅
intersecting ∂(B1 ) in an annulus. Let V = S3 \ int(N ). Then V is an unknotted
solid torus containing k. Note that V ∩ ∂(B1 ) is a disjoint union of two meridian
disks of V , one is intersecting k at x, and the other is intersecting k at 2n−1 points.
Then there is a homeomorphism h : V → S1 × D2 such that π ◦ h : V → S1 maps
each of these meridian disks to a point and each of k ∩ B1 and k ∩ B2 have exactly
n − 1 critical points of π ◦ f |k . Therefore f = [(V, k)] is a morphism from K to T
with mK (f ) ≤ 2n − 1. 
We now consider knot types K with m(K) ≤ n for n = 1, 2, 3, 4 as follows. Note
that the condition m(K) ≤ n is equivalent to the condition dmK (K, T ) ≤ loge 2n.
Proposition 5.10. Let K be an element of K.
(1) m(K) = 1 if and only if K is trivial.
(2) m(K) = 2 if and only if K is a (2, p)-torus knot for some odd number p 6= ±1.

MULTIPLICITY OF A SPACE OVER ANOTHER SPACE

21

(3) m(K) = 3 if and only if braid(K) = 3 or K is a connected sum of some 2-bridge
knots.
(4) If K is a Montesinos knot, then m(K) ≤ 4.
Proof. (1) It follows by Proposition 5.4 (1).
(2) It follows by Proposition 5.4 (2).
(3) Let f = [(V, k)] be a morphism from K to T with mK (f ) = 3. Let h : V →
S1 × D2 be a homeomorphism with m(h, k) = mK (f ) = 3. Suppose that there
are no critical points of π ◦ h|k . Then we have braid(K) ≤ 3. By Proposition
5.9 (1) braid(K) ≤ 2 implies m(K) ≤ 2. Therefore braid(K) = 3. Suppose that
there are some critical points of of π ◦ h|k . Then the local maximum and local
minimum must appear alternatively along S1 . Then we have a connected sum of
some 2-bridge knots. The converse is similar.
(4) For a Montesinos knot K it is easy to see m(K) ≤ 4. See for example Figure
5.3. 

Figure 5.3.

Acknowledgments
The author is most grateful to Professors Kazuaki Kobayashi, Shin’ichi Suzuki
and Ryuichi Ito. The author is also grateful to Professors Kazuhiro Kawamura,
Kazufumi Eto, Ken-ichiroh Kawasaki, Youngsik Huh, Makoto Ozawa and Ryo
Nikkuni for their helpful comments.
References
[1] W. Adkins and S. Weintraub, Algebra, An Approach via Module Theory, GTM 136, SpringerVerlag, 1992.
[2] S. Bogatyi, J. Fricke and E. Kudryavtseva, On multiplicity of mappings between surfaces,
Geom. Topol. Monogr., 14 (2008), 49–62.
[3] M. Gromov, Singularities, expanders and topology of maps, Part 2: From combinatorics to
topology via algebraic isoperimetry, Geom. Funct. Anal., 20 (2010), 416–526.
[4] K. Kanno and K. Taniyama, Braid presentation of spatial graphs, Tokyo J. Math., 33 (2010),
509–522.
[5] R. Karasev, Multiplicity of continuous maps between manifolds, preprint, arXiv:1002.0660.
[6] R. Nikkuni, Private communication, 2010.

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KOUKI TANIYAMA

[7] M. Ozawa, Waist and trunk of knots, Geom. Dedicata, 149 (2010), 85–94.
[8] D. Rolfsen, Knots and links, Publish or Perish, Inc., 1976.
[9] K. Taniyama, Multiplicity of a space over another space, Proceedings of Intelligence of Low
Dimensional Topology, (2007), 157-161.
[10] K. Taniyama, Multiplicity of a space over another space, Gakujutsu Kenkyu, School of Education, Waseda University, Series of Mathematics, 56 (2008), 1–4, in Japanese.
[11] K. Taniyama, Multiplicity distance of knots, RIMS Kokyuroku 1716, Intelligence of Lowdimensional Topology, (2010), 37–42.
Department of Mathematics, School of Education, Waseda University, Nishi-Waseda
1-6-1, Shinjuku-ku, Tokyo, 169-8050, Japan
E-mail address: taniyama@waseda.jp

