Tensor Products of Vector Bundles
We begin by reviewing vector bundles. An n-dimensional vector bundle is a
continuous map p (called the projection) from a space E (the total space) to the space B
(the base space), p : E  B , with a complex vector space structure n (where n is the
fiber dimension) on p 1  b  (the fiber over b) for each b  B on which we have local
trivializations . To define these local trivializations, we first cover B by open sets
U I such that for each U  , there exists a homeomorphism
h : p 1 U    U  
n
p 1  b  b  n
which is a vector space isomorphism for each b  B .
We can think of the local triviality condition as, in a small neighborhood of
b  B , the inverse image under p looks like the neighborhood with a copy of the vector
space attached at each point. Intuitively, as each h must be continuous, if we take two
points b and b’ in B that are close together, the fibers above b and b’ are also close
together in E and any twisting that occurs going from the fiber p 1  b  to the fiber
p1  b ' is insignificant.
If U i  U j   , we further require that the following diagram commutes:
This induces a homeomorphism
h j hi 1 : U i  U j   
 x, v 
where hi  x  : 
j
h 
j 1
i
n

i

n

j
n

i
 U i  U j   
 x , h  x  (v ) 
n

j
j
i
is a homeomorphism on the fibers above x such that
 hij on each fiber. Here, we have distinguished between the fibers above x  U i
and the fibers above x  U j , even though the fibers are isomorphic. The maps hi j are
called transition functions and, intuitively, tell us how the fibers are glued together.
Typically, we abuse terminology and refer to both the total space and the
projection p as the “vector bundle”. Also, we often identify the base space with its image
in the total space, don’t specify the trivializations, and don’t define the open cover of B.
The majority of texts referenced did not mention the transition functions at all, let alone
specifically. While discussing abuses of notation, we should mention the working idea
behind vector bundles; we build them as follows. Take a copy of the base space. Attach
a copy of the vector space to every point on the base space (this is our total space) in such
a way that the movement in the total space from one point to another can be done
smoothly. That means that the fibers must vary continuously over the base space.
Finally, the map p is the projects the every point in the fiber over b back down onto b.
Examples:
Line bundle over S1 Let p1 : E1  S 1 where each fiber is
1
i
two open sets U1 and U 2 . We want trivializations hi : p
. We cover the circle with
Ui   Ui 
. Choose V0 and
V1 disjoint open sets such that U 0 U1  V0 V1 . Define the gluing functions by
h1 h01 : U0 U1  
 b, v 
 U0 U1  
 b, v 

  b, v 
b  V1
b  V0
Mobius bundle over S1 Let p1 : E1  S 1 where each fiber is
. We cover the circle with
two open sets U1 and U 2 . We want trivializations hi : pi1 Ui   Ui 
. Choose V0 and
V1 disjoint open sets such that U 0 U1  V0 V1 . Define the gluing functions by
h1 h01 : U0 U1  
 b, v 
 U0 U1  
 b, v 

  b, v 
b  V1
b  V0
Note that the gluing functions define the difference between the bundles.
Now that we have vector bundles, we begin constructing new bundles from old
bundles. In class, we discussed the internal Whitney sum of vector bundles and subbundles. We can also form a sort of product of bundles, called the tensor product of
vector bundles, which is important in K-theory. Due to the involved nature of the
constructions of tensor products of vector bundles, we will look at three different
versions. The tensor product to be the end result of any of these constructions.
Construction 1:
Let p1 : E1  B and p2 : E2  B be vector bundles over the same base space B .
Consider the set defined by E1  E2 
bB
p11  b   p21  b  . For each open set U  B over
which E1 and E2 are trivial (that is, Ei  U 
1
i
hi : p
U   U 
ni
ni
), we choose isomorphisms
. We define a topology U on p11 U   p21 U  by requiring that
h1  h2 : p11 U   p21 U   U 
b  
p11  b   p21  b 

n1

n1

n2
n2


be a homeomorphism on the fibers. Now, suppose that we have another pair of
trivializations, ki : pi1 U   U  ni . Then
hi  pi1 U    U 
where the isomorphism in the center is
gi : U 
 b, v 
ni
U 
ni
ni
U 
 ki  pi1 U  
ni
 b, g b  v  
i
for each b U . Since the gi fix the first coordinate and map
them as gi : U  GLni 

ni
to
ni
, we can view
where the matrix might change over each fiber but changes
continuously, fiber to fiber. Thus, composition with this invertible matrix takes hi to k i .
Now, the entries of the matrices g1  b   g2  b  are the products of the entries of g1  b 
and g 2  b  . We can compose in an analogous fashion:
 h  b   h b    g b   g b    k b   k b  .
1
2
1
2
1
2
This means that the topology defined
on p11 U   p21 U  by requiring that k1  k2 be a homeomorphism on the fibers is the
same topology we get from requiring h1  h2 be a homeomorphism. The vector bundle
constructed this way is the tensor product of vector bundles E1 and E2 over B, denoted
E1 B E2 . This construction is a bit abstract for my taste, but does give the topology in a
nice fashion.
Construction 2:
Now, we look at a second construction that is a bit more intuitive. In order to do
so, we need to reconstruct the vector bundle as a quotient. Suppose we have vector a
bundle p : E  B and an open cover U I of B with local trivializations
h : p1 U   U 
n
. We can think of E as the disjoint union of the images of the
trivializations modulo an equivalence relation, E 
identifying U 
n
with U  
n

U


n

, where the relation
vb
is given by  b, v 
vb
h h1  b, v  whenever
b  U   U  . Since the base point, b, in U  is the same b in U  , the mapping
h h1 : U 
n
 U 
n
is essentially a mapping from
n
to another copy of
n
.
1
Thus, we can think of h h as assigning a matrix to each point b  U   U  that takes
the first copy of
n
into the second copy of
induced map g : U U   GLn 
1
1
.
n
. Thus, we identify h h1 with the
Note that these maps satisfy a cocycle condition
1
g g  h h h h  h h  g on U   U   U  . These are the gluing functions for
the vector bundle p : E  B .
Now, suppose we have p1 : E1  B (a vector bundle over B with fiber dimension
n1 ) and p2 : E2  B (a vector bundle over B with fiber dimension n2 ) as above, save that
we choose the open cover such that both bundles are trivial over the open cover. Then,
we have gluing functions g i : U  U   GLni   . We form the set
E1  E2 
bB
2
which
p11  b   p21  b  as above and choose gluing functions g1  g
2
assigns to each b  U   U  the tensor product of the matricies g1  b   g
b .
i
As g 
is induced by h h1 on each intersection, we have the local trivializations and the
necessary structure for the vector bundle E1  E2 .
Construction 3:
The third construction is considerably more complicated, but a bit more elegant.
We begin by defining the tensor product of two vector spaces. Let V be a vector space
over
with basis ei i 1 and W a vector space over
n
with basis  f j 
m
j 1
. Let


we make the following identifications:
V  W   ai , j  vi  w j  | vi V , w j W 
 i, j

 v  v '  w v  w  v '  w
v   w  w ' v  w  v  w '
r  v  w
 rv   w
v   rw
for all v, v ' V , w, w ' W , and r  . We call vi  w j a simple tensor. The space
V W is a vector space over
with basis ei  f j |1  i  n, 1  j  m .
Now, suppose that V’ and W’ are also vector spaces. Let  : V  V ' and
 : W  W ' be linear maps. We define
  : V  W  V '  W '
   v  w
  v    w
on simple tensors and extend linearly. Clearly, this is linear. Minimal work shows that it
is well defined.
We write the matrix for  in terms of our ordered basis for V and call it A. Likewise,
we write  in terms of our ordered basis for W and call it B. Then, we can write
A  B in terms of the basis for V W by multiplying each entry in A by B. This gives
an mn by mn matrix. For example, let
 b11 b12 b13 
 a11 a12 


A
B   b21 b22 b23 

 a21 a22 
b b

 31 32 b33 
Then,
 a11b11 a11b12 a11b13 a12b11 a12b12 a12b13 


 a11b21 a11b22 a11b23 a12b21 a12b22 a12b23 
 a B a12 B   a11b31 a11b32 a11b33 a12b31 a12b32 a12b33 
A  B   11


a
B
a
B
22 
 21
 a21b11 a21b12 a21b13 a22b11 a22b12 a22b13 
a b
a b
a21b23 a22b21 a22b22 a22b23 
 21 21 21 22

a
b
a
b
a
b
a
b
a
b
a
b
21 32
21 33
22 31
22 32
22 33 
 21 31
Incidentally, this tensor product of matrices is sometimes called the Kronecker product.
Next, we define a product more general than the tensor product and restrict to get
the product we want. Let X and Y be compact topological spaces. Let A  M n  C  X   and
B  M m  C Y   . Then, we define the external tensor product to be the matrix in


A  B  M nm  C  X  Y   defined by A  B  x, y   A  x   B  y  for all x  X , y  Y .
Now consider the vector bundles
p1 : V  X and p2 : W  Y where X and Y are
compact topological spaces. We write V  Ran  A and W  Ran  B  for A an
idempotent over X and B an idempotent over Y. Then, we define the external tensor
product of V and W to be the vector bundle V  W  Ran  A  Ran  B   Ran A  B

over X  Y . Note that if A and B are idempotents, so is A  B (via some matrix

multiplication) and that for each point  x, y   X  Y , the fiber V  W

x, y 
 Vx  Wy .
This is the tensor product of the vector spaces we discussed at the beginning of this
construction.
If we let X = Y for the vector spaces V and W, and let
: X  X  X
x  x, x 
be the diagonal map, we can define the (internal) tensor product of V and W by
V  W  * V  W . Since the diagonal map is continuous, and since the pullback of a



vector bundle by a continuous map is a vector bundle, the tensor product is also a vector
bundle.
After all of that, a picture or two might help with the intuition.
A Jumping Off Point for K-theory:
While the second construction is the most intuition (at least to me), the third leads
us to the following very quickly:
Corollary: Let X be a compact Hausdorff space. Then K 0  X  is a commutative ring
with unit under the internal tensor product.
To prove this, we look at a more general proposition:
Proposition: Let X and Y be compact Hausdorff spaces. The external tensor product
determines a group homomorphism
K 0  X   K 0 Y   K 0  X  Y 
Pf: Given differences of equivalence classes of vector bundles V   V '  K 0  X  and
W   W '  K 0 Y  , we define the product on simple tensors by
V   V '  W   W '  V  W   V  W '  V '  W   V '  W '
and then
extend by the distributive law to finite sums and constant multiples.
We need to show that this product is well defined. To do so, we first suppose
V   V '  V   V ' in K 0  X  . Recall that
V V ' V "  V V 'V "
for some vector bundle V " over X. Distributing the external tensor product across the
internal Whitney sum and recalling the definition of  E    F  , we have
V  W   V '  W   V "  W   V  W   V '  W   V "  W 
Thus,
V  W   V '  W   V  W   V '  W 

 
 
 

and
V  W   V '  W   V  W   V '  W  .

 
 
 

Hence, the product in K 0  X  is well defined. We make the same argument to show that
the product is well defined for representatives in K 0 Y  . This definition also respects
the defining relations for tensor products. The homomorphism is thus well defined.
Now, we restrict to the case Y = X and look at the corollary.
Corollary: Let X be a compact Hausdorff space. Then K 0  X  is a commutative ring
with unit under the internal tensor product.
Pf: Letting Y = X in the proposition, we have vector bundles over the same space. From
the third construction above, we know that the external tensor product becomes the
internal tensor product when we pullback by the diagonal map. The tensor product of
vector spaces is commutative and associative (see Dummit and Foote) and distributes
across internal Whitney sums. The previous theorem and the proposition stated below
imply that K 0  X  is a commutative ring with multiplicative identity element 1  X   ,
the trivial line bundle over X.
The proposition to which we alluded in the proof of the corollary is this:
Proposition: Let X and Y be compact Hausdorff spaces. Then the external tensor product
defines a map from Idem  C  X    Idem  C Y   to Idem  C  X  Y   .
For more information about any of these constructions, or if there are errors and you
would like to see a correct version, please see
Dummit, David and Foote, Richard. Abstract Algebra. Prentice Hall Publishing
Park, Efton. Complex Topological K-Theory. Cambridge UP. Print.
Hatcher, Allen. Vector Bundles and K-Theory. Copyright Allen Hatcher
Osborn, Howard. Vector Bundles: Foundations and Stieffel-Whitney Classes. Academic
Press, Inc. Print.
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