Questions tagged [index-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
1 answer
209 views

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 ...
truebaran's user avatar
  • 9,050
2 votes
0 answers
127 views

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 ...
zarathustra's user avatar
2 votes
0 answers
128 views

$\hat{A}$-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} =...
zarathustra's user avatar
3 votes
1 answer
163 views

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 ...
Yuji Tachikawa's user avatar
6 votes
0 answers
102 views

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 ...
Shaoyun Bai's user avatar
3 votes
0 answers
121 views

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 ...
Radeha Longa's user avatar
3 votes
0 answers
67 views

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 ...
Ho Man-Ho's user avatar
  • 971
2 votes
0 answers
56 views

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 ...
Mtheorist's user avatar
  • 1,105
3 votes
1 answer
102 views

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 ...
Overflowian's user avatar
  • 2,493
1 vote
0 answers
58 views

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 ...
Ho Man-Ho's user avatar
  • 971
6 votes
1 answer
239 views

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 ...
Matthew Niemiro's user avatar
5 votes
1 answer
227 views

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-...
ZZY's user avatar
  • 707
1 vote
0 answers
101 views

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 ...
Radeha Longa's user avatar
5 votes
0 answers
168 views

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,...
Isacu's user avatar
  • 51
9 votes
1 answer
601 views

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–...
Ya He's user avatar
  • 93
6 votes
0 answers
323 views

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 ...
dohmatob's user avatar
  • 6,466
2 votes
0 answers
210 views

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?)
user472816's user avatar
1 vote
0 answers
67 views

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 ...
Radeha Longa's user avatar
0 votes
1 answer
173 views

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 ...
Radeha Longa's user avatar
2 votes
0 answers
122 views

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(\...
D.Liu's user avatar
  • 21
4 votes
0 answers
169 views

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 ...
Ho Man-Ho's user avatar
  • 971
6 votes
0 answers
150 views

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 ...
Overflowian's user avatar
  • 2,493
9 votes
0 answers
510 views

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:...
Mahmoud Abdelrazek's user avatar
3 votes
0 answers
45 views

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 ...
geometricK's user avatar
  • 1,841
1 vote
0 answers
36 views

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 ...
geometricK's user avatar
  • 1,841
3 votes
0 answers
268 views

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 ...
annie marie cœur's user avatar
15 votes
1 answer
947 views

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 ...
Quaere Verum's user avatar
4 votes
0 answers
133 views

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 ...
BinAcker's user avatar
  • 747
2 votes
1 answer
243 views

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 $C_m(\tilde X)$ ...
Shiquan Ren's user avatar
  • 1,960
4 votes
0 answers
70 views

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) = \...
Tom's user avatar
  • 41
0 votes
1 answer
100 views

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 ...
YoYo's user avatar
  • 325
1 vote
0 answers
63 views

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} ( ...
YoYo's user avatar
  • 325
2 votes
0 answers
212 views

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 ...
gradstudent's user avatar
  • 2,136
5 votes
1 answer
229 views

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 ...
Rodrigo Dias's user avatar
8 votes
0 answers
386 views

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 ...
user267839's user avatar
  • 5,274
6 votes
1 answer
529 views

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 :...
user267839's user avatar
  • 5,274
6 votes
0 answers
287 views

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 ...
Gorapada Bera's user avatar
1 vote
0 answers
58 views

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 ...
Adnanne's user avatar
  • 11
2 votes
1 answer
135 views

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 ...
Liding Yao's user avatar
3 votes
0 answers
83 views

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 ...
Aleksandr Alekseev's user avatar
1 vote
0 answers
60 views

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 ...
DLIN's user avatar
  • 1,895
7 votes
0 answers
235 views

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|^...
FractalScout's user avatar
3 votes
0 answers
128 views

$\mathbb{Z}_2$-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 ...
Guest123412341234's user avatar
10 votes
1 answer
572 views

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 ...
Guest123412341234's user avatar
2 votes
0 answers
249 views

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, $$...
Pierre Dubois's user avatar
5 votes
1 answer
268 views

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 ...
12 votes
0 answers
517 views

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 \...
David Roberts's user avatar
  • 32.9k
0 votes
0 answers
61 views

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 $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$...
Ali Taghavi's user avatar
3 votes
1 answer
148 views

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 ...
Surpass2019's user avatar
0 votes
1 answer
93 views

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 ...
Max Schattman's user avatar