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/...
- 296
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 ...
- 43.2k
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What little background I have in this area is probably outdated, but I can share a few thoughts.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 ...
- 29.2k
20
votes
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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 ...
- 29.2k
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 ...
- 18.7k
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 ...
- 6,813
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}{...
- 34.7k
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:
$$\text{wind} f=\frac1{2\pi i}\int f'\bar{f}=\frac1{2\pi i}\sum\hat{f'}(n)\hat{f}(n)=\sum n\vert\hat{f}(n)\vert^...
- 43.2k
11
votes
Accepted
Is there a version of the Poincaré–Hopf theorem for manifold with corners?
With regards to the updated question: Note that the quoted statement is that the vector field points inward at the boundary. In particular this means that there are no singularities at the corners (...
- 2,504
11
votes
Accepted
Coefficient of the top Pontryagin class in $L$-genus
The coefficient of $p_k$ is given by
$$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 ...
- 8,167
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 ...
- 6,920
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 ...
- 1,338
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 ...
- 42.8k
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)$.
...
- 19.3k
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. ...
- 6,813
8
votes
Different proof techniques of the Atiyah-Singer index theorem
If you want something in lecture notes/book form, the book Stochastic Analysis on Manifolds by Elton Hsu contains a full exposition of the stochastic approach to the Atiyah-Singer index theorem, along ...
- 6,263
8
votes
Accepted
Is there a relationship between fusion and S^1-equivariance for spinors on loop space?
Matthias' paper that you cite is about the spinor bundle on loop spaces, and Matthias proves there that a manifold $M$ admits a string structure if and only if $LM$ admits a fusive spinor bundle.
The ...
- 4,519
7
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'...
- 19.4k
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}...
- 1,617
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 ...
- 29.2k
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 ...
- 29.2k
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 ...
- 3,451
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 ...
- 19.4k
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 ...
- 34.7k
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 ...
4
votes
Accepted
Why is index unchanged after applying functional calculus?
Perhaps the simplest answer is to use the spectral theorem: $L^2(S)$ decomposes as the orthogonal direct sum of $D$-eigenspaces, and $f(D)$ acts on each $\lambda$-eigenspace as multiplication by $f(\...
- 29.2k
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 ...
- 7,583
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
- 28k
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^...
- 5,343
4
votes
The index of certain differential operator on tori
$D$ is an elliptic operator, as its symbol is given by $$\sigma(D)(p,\xi)=(\xi(X_p))^2+(\xi(JX_p))^2$$ which is non-vanishing for $0\neq\xi\in T_pT^2.$ The index is always 0, as your operator can be ...
- 6,825
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