28
votes

Accepted

### Generalization of winding number to higher dimensions

This is a very naive answer which I am sure you already considered, but isn't the most obvious generalization just given by the topological degree (https://en.wikipedia.org/wiki/...

22
votes

Accepted

### Atiyah-Singer style index theorem for elliptic cohomology?

The status of this question is OPEN.
This theory has NOT been developed yet.
That being said, the evidence is as compelling as ever, I don't know of any obstructions to making this work, and I'm ...

22
votes

### Atiyah-Singer style index theorem for elliptic cohomology?

What little background I have in this area is probably outdated, but I can share a few thoughts. The "index theorem" to which Hopkins was most likely referring was in Witten's 1987 paper The Index of ...

18
votes

Accepted

### Atiyah-Singer theorem-a big picture

I agree with @coudy's answer that the best approach is to first understand the theorem's special cases / applications / generalizations. That can help highlight some of the key pain points in the ...

18
votes

### Atiyah-Singer theorem-a big picture

Well, I guess that there is no royal road to the Index Theorem.
I think that what is needed in order to understand the Atyiah-Singer index theorem is the opposite of a big picture. It is easier to ...

17
votes

### Atiyah-Singer theorem-a big picture

Just for completeness let me add a few words on the heat kernel proof. It is true that some analysis is needed. But one should not forget that even the $K$-theoretic proof depends on the notion of ...

15
votes

Accepted

### Atiyah's proof of the moduli space of SD irreducible YM connections

Hopefully I remember this well. My adviser explained this computation to me I don't even want to think how many years ago.
The deformation complex of the SD equation is $\DeclareMathOperator{\Ad}{...

14
votes

### Generalization of winding number to higher dimensions

A smooth function $f$ with image on the unit circle in $\mathbb{C}$ has winding number:
$$\text{wind} f=\frac1{2\pi i}\int f'\bar{f}=\frac1{2\pi i}\sum\hat{f'}(n)\hat{f}(n)=\sum n\vert\hat{f}(n)\vert^...

11
votes

Accepted

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

With regards to the updated question: Note that the quoted statement is that the vector field points inward at the boundary. In particular this means that there are no singularities at the corners (...

11
votes

Accepted

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

The coefficient of $p_k$ is given by
$$2^{2n}(2^{2n-1}-1)\frac{B_n}{(2n)!} = \zeta(2n)\frac{2^{2n}-2}{\pi^{2n}},$$
see e.g. Appendix A of this older version of Weiss (warning: for Weiss, the ...

10
votes

### Atiyah-Patodi-Singer Eta invariant and Chern-Simons form

Ad (3). Computing $\eta$-invariants on the nose is notoriously difficult. However, sometimes one needs $\eta$-invariants as ingredients of other differential topological invariants ($\rho$-invariants ...

10
votes

Accepted

### supersymmetry and the de Rham complex

I think there is a typo in the references to "Supersymmetry and Morse theory", [21] should be replaced by [22]="Constraints on supersymmetry breaking". The quantization of non-linear sigma models and ...

9
votes

### preliminary reading recommendation before embarking on Connes non commutative geometry book?

To understand everything in Connes' book you would need expertise in many different fields. My advice would be to browse it and see if anything attracts your interest. Then you can read up on the ...

9
votes

### Baum Connes conjecture and Atiyah-Singer index theorem

I find it overwhelmingly difficult to do this question proper justice. So instead I give a highly condensed answer and refer to Connes NCG, Section II.10 for more details (as well as all the ...

9
votes

Accepted

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

Your first question can be answered by using the splitting principle.
If $V \to X$ is a complex vector bundle of rank two, then $c_1(S^3V) = 6c_1(V)$ and $c_2(S^3V) = 11c_1(V)^2 + 10c_2(V)$.
...

8
votes

Accepted

### Index Theorem for the Twisted Dirac Operator

First, we need a spin structure to define the spinor bundle. The index theorem does not care which one we take, so we may take even spinors to be $(0,0)$-forms and odd spinors to be $(0,1)$ forms. ...

6
votes

### Injectivity/Surjectivity of $F_A :=\frac{d}{dt} +A(t) $ for a hyperbolic path of matrices $A(t)$ on $H^1 $

It is straightforward to see that if the path is constant then hyperbolicity is the equivalent to the invertibility of the operator for example via the Fourier transform (see Atiyah Patodi Singer (...

6
votes

Accepted

### How to compute the eta invariant of torus

The general formula can be found in Ouyang - Geometric Invariants For Seifert Fibred 3-Manifolds.
In particular, for $\Gamma \cong 1, \mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4, \mathbb{Z}_6, \mathbb{Z}...

6
votes

Accepted

### Coarse index of Dirac operator on $\mathbb{R}$

There are a number of ways to do this calculation, but at risk of shamelessly plugging my own work there is a nice way to see it using a Mayer-Vietoris principle.
Decompose $\mathbb{R}$ as the union ...

6
votes

Accepted

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

The assertion is supposed to be that $d(e^{-tD^2})/dt$ has the same smooth kernel as $-D^2 e^{-tD^2}$, i.e. they are the same operator. This is because $e^{-tD^2}$ is the solution operator to the ...

6
votes

Accepted

### can the actions of fundamental groups annihilate homology?

There are finitely presented groups that do not have any non-trivial linear representations, so for these groups as fundamental group you are just asking whether the ordinary real homology of $X$ is ...

6
votes

Accepted

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

Taken in the context of the introduction, I would guess that at least in part they are referring to applications of the index theorem to questions about positive scalar curvature. Before Lichnerowicz'...

5
votes

### Hirzebruch-Riemann-Roch theorem for Riemann surfaces with boundary

The Hodge-DeRham operator whose index on closed manifolds is the Euler characteristics admits local boundary conditions on manifolds with boundary, Dirichlet or Neumann type. That is not the case ...

5
votes

Accepted

### Index of Modified Dirac Operator

For this particular case the two operators are conjugate (though note the conjugacy is not unitary unless $s$ is imaginary)
$$
D_{f,s}=e^{-sf}D e^{sf}
$$
so that the dimension of both the kernel ...

5
votes

Accepted

### Dirac operator on manifold with periodic end

Not necessarily. The condition from Taubes's paper is stronger than just requiring the vanishing of the kernel of the Dirac operator $D_W$. Choosing $f:W \to S^1$ that is Poincaré dual to a multiple ...

5
votes

### Fredholm theory of non elliptic operators

I think I am in a good position to answer this. The Fredholm property of elliptic operators as maps between Sobolev spaces on compact manifolds rests on elliptic regularity properties. If an operators ...

Community wiki

4
votes

### Index of Modified Dirac Operator

Making my comment a formal answer:
The perturbation object is a compact operator (for any scaling $s$), and $D$ is Fredholm. The space of Fredholm operators is open in the (Banach) space of bounded ...

4
votes

### Is it possible to classify finite dimensional vector bundles in terms of Fredholm operators?

The problem with my remark from a couple of years ago is that the cokernel can also "move around". Let us therefore fix this.
Let $W\subset \mathbb{H}$ be a finite, say $l$, dimensional subspace of ...

4
votes

### Generalization of winding number to higher dimensions

The linking number is one of a natural generalizations of the winding number, see my answer to a related question: https://mathoverflow.net/a/297440/121665

4
votes

Accepted

### Elements of graded algebra associated with the algebra of differential operators as smooth sections

The vector space $U_x$ will be infinite dimensional, so it's not immediately clear what $\Gamma^\infty(M,U)$ denotes. I assume you mean $\Gamma^\infty(M,U):=\bigoplus_k \Gamma^\infty(M,U^k)$ where $U^...

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