Determinant line bundle

Definition

Let $H_0$ and $H_1$ be complex Hilbert spaces, and let $Y$ be a topological space. Let $y \mapsto T_y$ be a map from $Y$ to the Fredholm operators $H_0 \to H_1$ that is continuous in the operator norm. For a finite-dimensional space $V$ write $\det V = \Lambda^{\dim V} V$, with $\det 0 = \C$. The determinant line of $T_y$ is $$\det T_y = (\det \ker T_y)^* \otimes \det \operatorname{coker} T_y.$$ The determinant line bundle is the set $\det T = \bigsqcup_{y \in Y} \det T_y$ with the following topology. Near a point $y_0$, choose a finite-dimensional subspace $V \subseteq H_1$ with $\operatorname{im} T_y + V = H_1$ for all $y$ near $y_0$. Then the spaces $T_y^{-1}(V)$ form a vector bundle over a neighbourhood of $y_0$. The exact sequence $0 \to \ker T_y \to T_y^{-1}(V) \to V \to \operatorname{coker} T_y \to 0$ gives a canonical isomorphism $\det T_y \cong (\det T_y^{-1}(V))^* \otimes \det V$. The right side is a line bundle near $y_0$. The topology of $\det T$ is the one that makes these isomorphisms continuous, and it does not depend on the choice of $V$.