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-1
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0answers
39 views

A fredholm index associated with two vector fields generating a 2 dimensional foliation

Let $M$ be a compact manifold and $X,Y$ be two independent vector fields on $M$ with $[X,Y]=0$. Let $\mathcal{F}$ be the 2 dimensional foliation associated with the distribution ...
1
vote
1answer
129 views

How to extend index theorem to infinite dimensional manifolds?

I was thinking about the following "obvious" construction the other day. Let $$ \mathbb{S}^{\infty}_{2}=\bigcup_{i=1}^{\infty}\mathbb{S}^{2i} $$ Then because we know that $\chi(\mathbb{S}^{2i})=2$ for ...
2
votes
0answers
76 views

Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.It is a real analytic vector field on $S^{2}$ which ...
0
votes
0answers
68 views

Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...
2
votes
0answers
115 views

The space of Fredholm operators as a classifying space

Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces? Everyone quotes the theorem of Atiyah-Janich ...
0
votes
1answer
172 views

elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...
3
votes
0answers
100 views

How does the Atiyah-Singer index theorem in a relative setting related to “ringed spaces and pseudocoherent complexes of finite tor-dimension”?

I come across the following paragraph from the article Reminiscences of Grothendieck and His School, here is from the part of the interview by Luc Illusie,: " I was indeed looking for an ...
1
vote
0answers
86 views

Local Index Formula vs Atiyah Singer Index Theorem

I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue: why it is called Index Formula? I spoke to one person about this and he gave me the ...
7
votes
0answers
80 views

Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action. I am trying to understand the Hopf bundle ...
1
vote
1answer
160 views

Normal coordinates near the boundary

Let $M$ be an Riemannian manifold with boundary $\partial M$ and $e_n$ be a unit normal vector on $\partial M$. With respect to $e_n$, around a point $p$ on boundary, we have the usual normal ...
1
vote
1answer
193 views

How to classify continuous/differentiable maps from $T^2$ to $U(N)$?

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...
20
votes
2answers
605 views

Applications of Atiyah-Singer using pseudodifferential operators

Though the Atiyah-Singer index theorem holds for pseudodifferential operators, all the applications of the index theorem I know of only need it for Dirac-type operators. I know that pseudodifferential ...
11
votes
1answer
367 views

Atiyah-Singer for pseudodifferential operators via heat kernel?

The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
6
votes
1answer
323 views

What's the relation between the heat kernel proof of the index theorem and deformation quantization?

In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat ...
2
votes
1answer
191 views

How to compute the index of such operator?

Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of ...
1
vote
0answers
210 views

Non-elliptic deformation complex of some instanton-like equation

I have a very stupid and non-specific question, as follows. And let me know if I am asking in a wrong way. We known that, for instance, if one is interested in computing dimension of moduli space of ...
8
votes
1answer
181 views

Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition. What can ...
4
votes
1answer
213 views

Do we have a “topological assembly map” in the Baum-Connes conjecture?

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \text{a-ind}: ...
2
votes
1answer
223 views

The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold

This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...
14
votes
2answers
579 views

Karoubi versus Kasparov K-theory

I have the following, probably very elementary question: Let $Cl^{p,q}$ be the Clifford algebra on generators $e_i$, $i=1, \ldots, p+q$ with $e_i e_j = -e_j e_i$ and $e_{i}^{2}=-1$ for $i=1,\ldots,p$, ...
1
vote
1answer
226 views

Proof that the Hodge-de Rham Rank Equals the Euler Characteristic

Can someone please provide a good (online accessible) reference for the well-known identity $$ \text{rank((d + d}^*)^+) = \sum_{i=}^n (-1)^i \dim(H^i(M)), $$ where $M$ is a manifold of dimension $n$, ...
9
votes
0answers
432 views

What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
1
vote
1answer
210 views

Tricks to produce examples of hypersurfaces with index greater than $1$

Recall, index of an algebraic scheme $X$ is defined to be the greatest common divisor of the degrees of the space of zero cycles on $X$. I am interested in examples of hypersurfaces in ...
5
votes
1answer
200 views

Index theorems and orientability

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by ...
6
votes
1answer
313 views

Index of a differential operator between trivial bundles.

Let $M$ be a closed parallelizable manifold and $D: \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between trivial vector bundles $E,F \to M$. The Atiyah Singer index theorem implies that ...
2
votes
1answer
275 views

Kasparov's Dirac element and the index map

In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology ...
3
votes
0answers
172 views

The proof of the splitting principle in equivariant K-theory via flag manifolds

In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely: Let $j: ...
5
votes
0answers
164 views

Relative index theorem for Clifford linear Dirac operators

Dear community, there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that ...
1
vote
0answers
105 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
4
votes
1answer
613 views

The equivariant index of Dirac operator

Let us consider the Dirac complex \begin{equation} D_{\rm Dirac}:S^+\to S^- \end{equation} where $S^{\pm}$ are the chiral-spinor bundles on $\mathbb{R}^4$. Using the fact that the bundle $S^+$ is ...
7
votes
0answers
267 views

Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?

I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner. They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
14
votes
3answers
782 views

Geometric meaning of L-genus

Is there any reasonable geometric meaning of the L-genus for smooth manifolds? or perhaps easier, for complex algebraic surfaces? The question came up after a friend and I realized that we don't ...
15
votes
2answers
644 views

What is the role of equivariance in the Atiyah-Singer index theorem?

I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof: Define (using purely K-theoretic means) a homomorphism $K_G(TX) \to ...
1
vote
1answer
380 views

If V is an irreducible representation of G, what is K_{G}(T_{G}V)?

Here, G is a compact lie group. V is a finite dimensional irrepn of G. By Atiyah, every element in K_{G}(T_{G}V) is a symbol of a transversally elliptic operator on V. Of course, K_{G}(T_{G}V) is ...
6
votes
0answers
261 views

Between Being a Connection and Being an Elliptic Operator

Let $E$ be a smooth complex vector bundle over a smooth compact manifold $M$ and let $H$ be the $Z_{2}$-graded (with the $Z_{2}$-grading given by even/odd forms) Hilbert space of $L_{2}$ (with respect ...
6
votes
2answers
676 views

Understanding the analytic index map of the Atiyah-Singer index theorem

Hi, I'm currently trying to understand the Atiyah-Singer index theorem and its proof as presented in the book "Spin Geometry" by Lawson and Michelsohn. I do not understand why the analytic index map ...
2
votes
1answer
315 views

Index of elliptic operators III: H-structure invariant under a group G

In the Atiyah-Singer paper mentioned above, they introduced on p.557 a concept called $H$-structure which is used to describe the Chern character of special elements of $K(TX)$. It is roughly the ...
1
vote
1answer
320 views

In K-homology K(X), if the Dirac operator D is invertible, does [D] represent zero element?

When X = pts, we know that the index of [D] equal to 0. What about X is not a point. Thanks