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### 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 ...

**15**

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**3**answers

893 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**

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**3**answers

2k views

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

I am trying to understand the Atiyah-Patodi-Singer index theorem in the case of Dirac operators in four dimensions. I have three questions about the eta invariant:
1) Is eta a topological invariant (...

**16**

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**2**answers

742 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 R(G)...

**1**

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**1**answer

399 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 ...

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**0**answers

304 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 ...

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**2**answers

748 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

**1**answer

320 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**

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**1**answer

364 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

**60**

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**3**answers

6k views

### Intuitive explanation for the Atiyah-Singer index theorem

My question is related to the question Explanation for the Chern Character to this question about Todd classes, and to this question about the Atiyah-Singer index theorem.
I'm trying to learn the ...

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**8**answers

9k views

### What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...