Loop space of Fredholm operators from a Relative loop space

Question

Atiyah and Singer proved that the nontrivial component of the set of skew-adjoint Fredholm operators $ \hat{\mathcal{F}_{*}}(\mathscr{H})$ is homotopic to the loop space of Fredholm operators $\Omega\mathcal{F}(\mathscr{H})$.

But in their paper and in other sources what they prove is homotopy to a so-called relative loop space $\Omega(\mathcal{L},\mathcal{-C})$ where $\mathcal{L}$ is a retraction of the group of units of $\mathcal{B}(\mathscr{H})$, the set of bounded operators of a complex separable Hilbert space $\mathscr{H}$ and $\mathcal{-C}$ are operators of the form $I - T$, where $T$ is compact.

1) Does anyone know how to pass from the relative loop space $\Omega(\mathcal{L},\mathcal{-C})$ to $\Omega\mathcal{F}(\mathscr{H})$?

A related question is the following: By Atiyah-Singer $\mathbb{Z}\approx\mathcal{K}^{-1}(point) \approx [point,\hat{\mathcal{F}_{*}}(\mathscr{H})]$

2) Since $\forall A(t) \in \hat{\mathcal{F}_{*}}(\mathscr{H}), (0\leq t\leq 1)$, $Ind\, A(t) = 0 \,\,\forall t$ and $Ker\,A(t) \in \mathbb{Z}^{+} \,\,\forall t$, what property of $A(t)$ determines the equivalence class it belongs to in $\mathbb{Z}$?

Thank you for reading

Answer 1

On a complex Hilbert space $H$ we can easily identify the space of skew-adjoint operators with the space of self-adjoint operators. (Multiplication by $\sqrt{-1}$ will do the trick. The space $\newcommand{\FS}{\mathscr{FS}}$ $\FS$of Fredholm selfadjoint operators on $H$ has three components $\FS_{\pm}$, $\FS_*$. The first two are homotopically trivial, the third $\FS_*$ classifies $K^1$. In particular, $\newcommand{\bZ}{\mathbb{Z}}$ $\pi_1(\FS_*)\cong \bZ$ and there is a canonical isomorphism $\pi_1(\FS_*)\to\bZ$ called the spectral spectral flow. You can visualize this isomorphism as follows.

The space $\FS_*$ is homotopy equivalent with the infinite dimensional Banach manifold $\newcommand{\L}{\mathscr{L}}$ $\L(H)$ consisting of the subspaces $L\subset H\oplus H$ satisfying the conditions

• The pair $(L, H\oplus 0)$ is a Fredholm pair, i.e. $\dim L\cap (H\oplus 0)<\infty$ and $L+(H\oplus 0)$ is closed in $H\oplus H$.
• The subspaces $L$ is Lagrangian, i.e., $L^\perp=JL$, where $J:H\oplus H\to H\oplus H$ is defined by $J(x,y)=(-y,x)$.

There is a natural map $\FS(H)\to \L(H)$ that associated to the operator $T$ its graph $\Gamma_T\subset H\oplus H$. This map is a homotopy equivalence.

As I mentioned above, $\L(H)$ is a Banach manifold and, moreover, it is equipped with a Schubert-like stratification by smooth strata with finite codimension. This stratification contains a single codimension-1 stratum $\newcommand{\M}{\mathscr{M}}$ $\M$ that is naturally co-oriented. (This stratum is sometime called the Maslov divisor.) Given smooth map $\alpha:S^1\to \L(H)$ we denote by $\mu(\alpha)$ its intersection number with $\M$. The resulting map $\mu:\pi_1(\L(H))$\to\bZ$ is an isomorphism. There are similar descriptions for the isomorphisms

$$\pi_{2k-1}(\L(H))\to\bZ,\;\;k=1,2,\dotsc. $$

For details, I refer to Daniel Cibotaru's dissertation.