# Questions tagged [index-theory]

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### Comments and reference-request on books for KK-theory

I am looking for a good reference to learn Kasparov's KK-theory, where my motivation is to understand (and hopefully can do something about) the Atiyah-Singer index theorem in terms of KK-theory. I ...
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### Elliptic operators with with same index but non homotopic symbols

Let $\mathcal{D}:\Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $k$. Where $E,F$ are $\mathbb{C}$-vector bundles over $X$, a compact smooth manifold. In Atiyah-Singer "the index of ...
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### Why is the symbol map in Atiyah–Singer paper continuous?

I am reading "Index of elliptic operators: I", by Atiyah and Singer these days and I am trying to understand all the paper. I find it difficult to verify the following statement on page 512:...
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### Dependence of Roe algebra and coarse index on the Riemannian metric

Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
1 vote
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### Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
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### Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete. For example at least for $d=4$, this ...
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### Atiyah's proof of the moduli space of SD irreducible YM connections

In the paper "Self-duality in Four-dimensional Riemannian Geometry" (1978), Atiyah, Hitchin and Singer present a proof that the space of self-dual irreducible Yang-Mills connections is a ...
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### Push forward of Chern character and index theorem

I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998). I expose here the setup for my ...
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### can the actions of fundamental groups annihilate homology?

Let $X$ be a path-connected manifold (or a CW complex). Let $\pi_1(X)$ be the fundamental group of $X$. Let $\pi: \tilde X\longrightarrow X$ be a covering map. For each $m\geq 0$, let $C_m(\tilde X)$ ...
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### Spectral flow of Dirac operator twisted by instanton

Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
1 vote
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### Splitting formulas for spectral flows

I'm asking if there are splitting formulas for equivariant spectral flow and higher spectral flow (of Dai-Zhang) for paths of Dirac operators, concerning gluing together two smooth compact Spinc ...
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### Is being a deg 0 vector field equivalent to being locally non-surjective?

Let $X:\mathbb R^n\to\mathbb R^n$ be a $C^1$ (or smooth)-vector field, such that $X(0)=0$ is an isolated zero. So we can talk about the mapping of $0$ for $X$. For convenience assume $0$ be the only ...
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### Dixmier trace, Wodzicki residue and topological index

There are a well-known facts about Dixmier trace and Wodzicki residue. Let $P$ be an elliptic pseudodifferential operator of degree $−n$ on a compact Riemannian manifold $(M,g)$, than its Dixmier ...
1 vote
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### About the orientation of index formula on orbifold

Let $X$ be a closed oriented orbifold with singularity $\Sigma X$. The singularity is defined as $$\Sigma X=\{(x)|~x\in X,~G_x\neq1\},$$ where $G_x$ is the isotropy group. For $u\in K_v(TX)$, as ...
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### Fredholm theory of non elliptic operators

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in ...
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### Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields. For example, the first Chern class of a complex line ...
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### Matrix of cosecants appearing in equivariant index computations

In a computation of characters of certain representations of finite cyclic groups which appear as equivariant indices of Dirac operators (using the Atiyah-Bott fixed point formula, cf. [1, Theorem 8....
### Producing $K$-homology cycles from $KK$-cycles
For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :) I wonder if there us a natural way to "forget" the ...