# Questions tagged [index-theory]

The index-theory tag has no usage guidance.

165
questions

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### Determinant line of Fredholm operators and composition of morphisms

Let $P$ be a polarization of a Hilbert space $\mathcal{H}$, i.e. a bounded idempotent: consider a group $G=GL_{res}(\mathcal{H}):=\{g \in GL(\mathcal{H}): [g,P] \in HS\}$ (where $HS$ is the set of all ...

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127
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### Proof of the Hirzebruch-Riemann-Roch theorem using the Atiyah-Singer index theorem

I am trying to read the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), but I do not understand the few last steps (theorem 4.11, page ...

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128
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### $\hat{A}$-genus of a complex manifold

I am trying to understand the proof of the Riemann-Roch-Hirzebruch theorem using the index theorem (Heat Kernel and Dirac operators, [BGV]), and at the end they say that since $$TM \otimes \mathbb{C} =...

3
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### Explicit computations of Bismut-Cheeger eta form for $S^{2n}$ bundles

I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the ...

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### K-homology fundamental class for singular varieties?

Given a smooth $\text{Spin}^c$ compact manifold without boundary $M$, a suitable normalization of the Dirac operator defines the fundamental class of $M$ in Kasparov's $KK(\mathbb{C}, C^0(M))$. This ...

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121
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### A question about index of Dirac operator

Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...

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### A confusion about an assumption in the setting of the local family index theorem

Let $\pi:M\to B$ be a proper submersion with closed, oriented and spin fibers. Then one can state the local family index theorem as an equality of differential forms (ignoring the details here).
I am ...

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### Size of Hilbert space in geometric quantization from index theorem

In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem.
To be precise, the polarization ...

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1
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102
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### Index formula for elliptic operators acting on Sobolev sections vanishing on the boundary (say $D: H_0^k(\Omega) \to H_0^{k-1}(\Omega)$)

Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the ...

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58
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### A proof that the analytic index for families is multiplicative

I am looking for a detailed proof of the analytic index for families (of geometrically defined Dirac operators is good enough and assuming the existence of the kernel bundle) is multiplicative. Any ...

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239
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### What were the "questions unapproachable by other means" w.r.t. $KO$-invariants?

H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin Geometry, (1989), p. xi:
...This formula was to generalize the important [HRR]. ...Atiyah and Singer...produced a globally defined elliptic ...

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227
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### Coefficient of the top Pontryagin class in $L$-genus

The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows:
$$L_1=\frac{1}{3}p_1,$$
$$L_2=\frac{1}{45}(7p_2-p_1^2),$$
$$L_3=\frac{1}{945}(62p_3-...

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101
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### Existence of a local spinor bundle

I am confused about the existence of a local spinor bundle.
My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open ...

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168
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### Was an index theorem for manifold with local boundary condition proven?

I would like to ask a question on the bibliography of the index theorems on manifold with boundary.
Before my bibliographical research my understanding of the field was that for manifold with boundary,...

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601
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### Is there a version of the Poincaré–Hopf theorem for manifold with corners?

As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...

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323
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### Atiyah–Singer Index theorem for the pedestrian / layperson

So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool.
Question. What is a truly simple application of the ASIT to obtain a ...

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### Riemann-Roch theorem for higher-dimensional complex manifolds

Does an analogue of the Riemann-Roch theorem hold for higher-dimensional complex manifolds? (Hirzebruch-Riemann-Roch theorem is for algebraic manifolds, but not for general complex manifolds, right?)

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### Relationship with between Clifford multiplication and pullback

Let $X$ be a smooth vector field on the even-dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...

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173
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### Question about Clifford multiplication

Let $X$ be a smooth vector field on the even dimensional sphere $S^n$. Let $S(TS^n)=S^+(TS^n)\oplus S^-(TS^n)$ be the spinor bundle over $S^n$ equipped with a bundle metric that is compatible with the ...

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122
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### Friedrichs Inequality

I'm a little confused with the following proof of Friedrichs inequality in Lawson's & Michelsohn's book Spin geometry, page 194, Theorem 5.4.
I don't understand why the last inequality, i.e.
$$
C(\...

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169
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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 ...

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36
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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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268
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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 ...

15
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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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133
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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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1
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243
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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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### Analyticity of the regularized $\eta$-invariant

The APS $\eta$- invariant of an operator $B$ with eigenvalues $\lambda$ is defined as
$$\eta = \sum_\lambda sgn (\lambda)$$
which is a divergent sum and it can be regularized as follows:
$$\eta(s) = \...

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1
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100
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### How to define an equivariant Kasparov's KK-theory map?

I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...

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### A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov

Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...

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### Spectrum of the Witten Laplacian on compact Riemannian manifolds

Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...

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### Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.
For a continuous ...

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### Atiyah-Singer theorem in heat kernels and Dirac operators

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne.
I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the ...

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529
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### McKean-Singer formula in Heat Kernels and Dirac Operators book

I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$
and $D :...

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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 ...

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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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135
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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 ...

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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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### Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$

The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by
$$
\eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...

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### $\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)

This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...

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572
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### Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...

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249
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### Twisting holomorphic vector bundles and Euler characteristics

Given a holomorphic vector bundle $\mathcal{V}$ over a compact complex manifold $M$, it seems that even if $\mathcal{V}$ is non-trivial, then it can still have trivial Euler characteristic, that it,
$$...

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268
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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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### Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?

The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...

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61
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### A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...

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148
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### Is there a fixed-point index theorem that treats the fixed points on the boundary?

Suppose $f$ is a continuous function from a unit cube $[0,1]^n$ to itself, then $f$ has at least a fixed point. Further suppose $f$ is smooth, $0$ is a regular value of $f(x)-x$, and the fixed points ...

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93
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### An adjoint characterization of (unbounded) Fredholm operators

Let $\mathcal{H}_i$, for $i=1,2$, be Hilbert spaces, and $T:{\frak Dom}(T) \subseteq \mathcal{H}_1 \to \mathcal{H}_2$ a densely-defined closed operator. If the kernel of $T$, and the kernel of its ...