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CONSTRUCTION

The Total Space EU(n) of the Universal Bundle is given by

:EU(n)=\{e_1,\ldots,e_n : (e_i,e_j)=\delta_{ij}, e_i\in \mathcal{H} \}.

Here, ''H'' is an infinite-dimensional complex Hilbert space, the e_i are vectors in ''H'', and \delta_{ij} is the Kronecker Delta . The symbol (\cdot,\cdot) is the Inner Product on ''H''. Thus, we have that ''EU(n)'' is the space of Orthonormal ''n''-frames in ''H''.

The Group Action of ''U''(''n'') on this space is the natural one. The Base Space is then

:BU(n)=EU(n)/U(n)

and is the set of Grassmannian ''n''-dimensional subspaces (or ''n''-planes) in ''H''. That is,

:BU(n) = \{ V \subset \mathcal{H} : \dim V = n \}

so that ''V'' is an ''n''-dimensional vector space.


VALIDITY OF THE CONSTRUCTION

In this section, we will define the topology on ''EU(n)'' and prove that ''EU(n)'' is indeed contractible.

Let F_n(\mathbb{C}^k) be the space of orthonormal families of n vectors in \mathbb{C}^k. The group U(n) acts
freely on F_n(\mathbb{C}^k) and the quotient is the Grassmannian G_n(\mathbb{C}^k) of n-dimensional subvector spaces of \mathbb{C}^k. The map

: \begin{align}
F_n(\mathbb{C}^k) & \longrightarrow & S^{2k-1} \
(e_1,\ldots,e_n) & \longmapsto & e_n
\end{align}

is a fibre bundle of fibre F_{n-1}(\mathbb{C}^{k-1}). Thus because \pi_p(S^{2k-1}) is trivial and because of the Long Exact Sequence Of The Fibration , we have

: \pi_p(F_n(\mathbb{C}^k))=\pi_p(F_{n-1}(\mathbb{C}^{k-1}))

whenever p\leq 2k-2. By taking k big enough, precisely for k> rac{1}{2}p+n-1, we can repeat the process and get

: \pi_p(F_n(\mathbb{C}^k)) = \pi_p(F_{n-1}(\mathbb{C}^{k-1})) = \cdots = \pi_p(F_1(\mathbb{C}^{k+1-n})) = \pi_p(S^{k-n}).

This last group is trivial for k>n+p. Let

: EU(n)={\lim_{ ightarrow}}\;_{k ightarrow\infty}F_n(\mathbb{C}^k)

be the Direct Limit of all the F_n(\mathbb{C}^k) (with the induced topology). Let

: G_n(\mathbb{C}^\infty)={\lim_{ ightarrow}}\;_{k ightarrow\infty}G_n(\mathbb{C}^k)

be the Direct Limit of all the G_n(\mathbb{C}^k) (with the induced topology).

Lemma

The group \pi_p(EU(n)) is trivial for all p\ge 1.

Proof
Let \gamma be a map from the sphere S^p to ''EU(n)''. As S^p is Compact ,
there exists k such that \gamma(S^p) is included in F_n(\mathbb{C}^k). By taking k big enough,
we see that \gamma is homotopic, with respect to the base point, to the constant map.
\Box

In addition, U(n) acts freely on ''EU(n)''. The spaces F_n(\mathbb{C}^k) and G_n(\mathbb{C}^k) are CW-complexes . One can
find a decomposition of these spaces into CW-complexes such that the decomposition of F_n(\mathbb{C}^k), resp.
G_n(\mathbb{C}^k), is induced by restriction of the one for F_n(\mathbb{C}^{k+1}), resp. G_n(\mathbb{C}^{k+1}). Thus ''EU(n)'' (and also G_n(\mathbb{C}^\infty)) is a CW-complexe. By
Whitehead Theorem and the above Lemma, ''EU''(''n'') is contractible.


CASE OF ''N'' = 1


In the case of ''n'' = 1, one has

:EU(1)= \mathbb {C}^\infty.\,

Taking the quotient of \mathbb{C}^\infty by an action of \mathbb R^+, the group of positive numbers by multiplication (this does not change the homotopy type of the space, \mathbb R^+ being isomorphic to R), one sees that the space is essentially a unit ball in a complex countable-dimension vector space. The base space is then

:BU(1)= \mathbb{C}P^\infty,\,

the infinite-dimensional Complex Projective Space . Thus, the set of Isomorphism Class es of Circle Bundle s over a Manifold ''M'' are in one-to-one correspondence with the Homotopy Class es of maps from ''M'' to \mathbb{C}P^\infty.

One also has the relation that

:BU(1)= PU(\mathcal{H}),

that is, BU(1) is the infinite-dimensional Projective Unitary Group . See that article for additional discussion and properties.


COHOMOLOGY OF THE CLASSIFYING SPACE BU(N)

Proposition

  • (BU(n)) is a Ring of Polynomials in ''n'' variables

    • c_1,\ldots,c_n where c_p is of degree ''2p''.


Proof
Let us first consider the case ''n=1''. In this case, ''U(1)'' is the circle S^1 and the universal bundle
is S^\infty\longrightarrow CP^\infty. It is well knownR. Bott, L. W. Tu
-- ''Differential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82,
Springer that the cohomology of
CP^k is isomorphic to \mathbb{R}\lbrack c_1 brack/c_1^{k+1}, where c_1 is the Euler Class of
the ''U(1)''-bundle S^{2k+1}\longrightarrow CP^k, and that the injections CP^k\longrightarrow CP^{k+1},
  • , are compatible with these presentations of the cohomology of the projective spaces.

    • This proves the Proposition for ''n=1''.


In the general case, let ''T'' be the subgroup of diagonal matrices. It is a Maximal Torus in ''U(n)''. Its
classifying space is (CP^\infty)^n and its cohomology is \mathbb{R}\lbrack x_1,\ldots,x_n brack, where
x_i is the `Euler class' of the tautological bundle over the ''i''-th CP^\infty. The
Weyl Group acts on T by permuting the diagonal entries, hence it acts on (CP^\infty)^n by
permutation of the factors. The induce action on its cohomology is the permutation of the
x_i's. We deduce
  • (BU(n))=\mathbb{R}\lbrack c_1,\ldots,c_n brack,

    • where the c_i's are the Symmetric Polynomials in the x_i's.

\Box



SEE ALSO



REFERENCES