noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
A continuous linear operator $F \colon B_1\to B_2$ between Banach spaces is Fredholm if it has finite dimensional kernel and finite dimensional cokernel.
The difference between the dimensions of the kernel and the cokernel of a Fredholm operator $F$ is called its index
A standard equivalent characterization of Fredholm operators is the following:
A parametrix of a bounded linear operator $F \colon \mathcal{H}_1 \to \mathcal{H}_2$ is a reverse operator $P \colon \mathcal{H}_2 \to \mathcal{H}_1$ which is an “inverse up to compact operators”, i.e. such that $F \circ P - id_{\mathcal{H}_2}$ and $P \circ F - id_{\mathcal{H}_1}$ are both compact operators.
A bounded linear operator $F \colon B_1\to B_2$ between Banach spaces is Fredholm, def. 1 precisely it is has a parametrix, def. 3.
The image (range) of a Fredholm operator is closed.
The subspace $Fred(B_1,B_2)\subset B(B_1,B_2)$ of Fredholm operators in the space of bounded linear operators with the norm topology is open.
In other words, given a Fredholm operator $F$, there exists $\epsilon\gt 0$ such that every bounded linear operator $G$ satisfying $\| G-F\|\lt \epsilon$ is Fredholm. Fredholm operators on a fixed separable Hilbert space $H = B_1 = B_2$ form a semigroup with respect to the composition and the index is a morphism of semigroups: $ind F G = ind F + ind G$.
The space $Fred$ of all Fredholm operators on an (infinite dimensional) separable Hilbert space is a model for the classifying space of degree-0 topological K-theory.
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Fredholm operators generalize to Fredholm complexes. A finite chain complex
of Banach spaces and bounded operators is said to be a Fredholm complex if the images $d_i$ are closed and the chain homology of the complex is finite dimensional. The Euler characteristic (the alternative sum of the dimensions of the homology groups) is then called the index of the Fredholm complex. Each Fredholm operator can be considered as a Fredholm complex concentrated at zero. Each Fredholm complex produces a Fredholm operator from the sum of the even- to the sum of the odd-numbered spaces in the complex.
One can consider Fredholm almost complexes, where $d_i \circ d_{i-1}$ is not zero but the image of that operator is compact. Out of every Fredholm almost complex one can make a complex by correcting the differentials by compact perturbation terms. Elliptic complexes give examples of Fredholm complexes.
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