# A curious switch in infinite dimensions

Let $$V$$ be a finite dimensional real vector space. Let $$GL(V)$$ be the set of invertible linear transformations, and $$\Phi(V)$$ be the set of all linear transformations. We can also characterize $$\Phi(V)$$ as those linear transformations with a finite dimensional kernel and a finite dimensional cokernel, i.e. these are exactly the Fredholm operators in finite dimensions.

If $$\mathbb{H}$$ denotes an infinite dimensional separable Hilbert space, we can also consider the space of all bounded linear transformations $$GL(\mathbb{H})$$, and the set of all bounded linear transformations which have finite dimensional kernel and cokernel. The latter is the space of (bounded) Fredholm operators $$\Phi(\mathbb{H})$$.

Many invariants in finite dimensional geometric topology use, either implicitly or explicitly, the fact that $$GL(V)$$ has non-trivial topology. For example, it is the reason that vector bundles can be non-trivial. In contrast, $$\Phi(V)$$ is contractible.

In infinite dimensional topology all these geometric invariants vanish. Kuiper's theorem states that $$GL(\mathbb H)$$ is contractible, and a corollary is that all infinite dimensional (seperable Hilbert) vector bundles are trivial. But, the space of Fredholm operators $$\Phi(\mathbb{H})$$, which I have introduced as the analogon of $$\Phi(V)$$ in infinite dimensions, is not contractible. Most invariants that I know in infinite dimensional topology actually use the fact that the homotopy type of $$\Phi(\mathbb H)$$ is non-trivial.

By the Atiyah-Jänich theorem $$\Phi(\mathbb{H})$$ classifies topological $$KO$$-theory and therefore the homotopy type of $$\Phi(\mathbb{H})$$ is closely related to the homotopy type of the classifying space $$BGL(V)$$, when the dimension of $$V$$ is large. Namely, take a sequence of finite dimensional vector spaces $$V_i$$ of increasing dimension, along with compatible inclusions. Then we can define the limiting group $$GL(V_\infty):=\lim_{i\rightarrow \infty} GL(V_i)$$. Then the homotopy type of $$\Phi(\mathbb{H})$$ is then that of $$BGL(V_\infty)\times \mathbb{Z}$$.

Now the classifying space of $$GL(\mathbb{H})$$ has the homotopy type of a point. Any group acts freely on itself, and in this case the group is contractible. The quotient is just a point, but also a model of $$BGL(\mathbb{H})$$.

Hence the topology $$BGL(V)$$ resembles that of $$\Phi(\mathbb{H})$$ for $$V$$ sufficiently large, and the topology of $$\Phi(V)$$ resembles that of $$BGL(\mathbb H)$$.

Now I think I understand the proofs of these facts, but I still find it miraculous that there is this "switch" when passing to infinite dimensions. My question is if there is some big picture reason why one should expect this.

• I guess I would say: $\Phi(V)$ is completely unconstrained when $V$ is finite-dimensional, while it is constrained by finite-dimensionality once you reach infinite dimensions, and the interesting topology accounts for how kernel and cokernel move around inside an infinite-dimensional space. On the other hand, when $V$ is infinite-dimensional the space $GL(V)$ is unconstrained because of the existence of a swindle. – Mike Miller Aug 17 at 20:16
• My vague memory is that this information is better organized in terms of Grassmannians than groups, but I don't have anything specific to say. – Ben Wieland Sep 17 at 16:44