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Ho Man-Ho
Ho Man-Ho
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The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the underlying setting of my questions is given by the local family index theorem with odd dimensional fibers. My questions are somehow related to the post "https://mathoverflow.net/questions/43175/index-of-a-family-of-dirac-operators-in-k1".

The background of my questions is the following. Let $\pi:X\to B$ be a proper submersion whose fibers are odd dimensional oriented spin manifolds and $E\to X$ a complex vector bundle. After putting the necessary geometric data (metrics, horizontal distribution, various unitary connections), we have a spin Dirac operator $D$. To define the analytic index $\textrm{ind}(D)\in K^{-1}(B)$, we define an operator $D_\theta$ following (3.2) of page 81 of Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry III". Then we get an index $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$. Since its restriction to $\{0\}\times B$ is trivial, it lies in $\ker(i^*:K^0(\mathbb{S}^1\times B)\to K^0(B))\cong K^{-1}(B)$, where $i:B\to\mathcal{S}^1\times B$ is the inclusion map, and $\textrm{ind}(D)\in K^{-1}(B)$ is defined to be $$\textrm{ind}(D)=\textrm{ind}(D_\theta)\in K^{-1}(B).$$ More precisely, the right-hand side of the above should be regarded as the corresponding element.

My questions are

  1. If, by some luck, the family of kernels $\ker(D_b)$ form a vector bundle (which is ungraded), does $\ker(D)\to B$ represent $\textrm{ind}(D)\in K^{-1}(B)$?

  2. If the answer to Q.1 is yes, what can we say about $\ker(D)\to B$ and $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$? Or more precisely, if $\ker(D)\to B$ exists, does $\ker(D_\theta)\to B$ exist? If so, how are they related?

  3. If the answer to Q.1 is no, is the "suspension trick" the only way to defineLet's suppose $\textrm{ind}(D)$?$\ker(D_b)$ do not form a bundle. It's well known that in the even-dimensional fiber case, if $\ker(D)\to B$$\ker(D_\theta)\to B$ does not exist one can perturb $D$$D_\theta$ so that $\textrm{ind}(D)=[\ker(D')]-[\mathbb{C}^n]$$\textrm{ind}(D_\theta)=[\ker(D_\theta')]-[\mathbb{C}^n]$, where $D'$$D_\theta'$ is the perturbed operator and $[\mathbb{C}^n]$ means it is represented by a trivial bundle of rank $n$ for some $n$. Is there any "perturbation method" in the odd dimensional fiber case? So I meant, is there any way to define $\textrm{ind}(D)\in K^{-1}(B)$ directly even though $\ker(D_b)$ do not form a bundle (this kind of implicitly assumes that the answer to Q.1 is yes) without going through the "suspension trick"?

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the underlying setting of my questions is given by the local family index theorem with odd dimensional fibers. My questions are somehow related to the post "https://mathoverflow.net/questions/43175/index-of-a-family-of-dirac-operators-in-k1".

The background of my questions is the following. Let $\pi:X\to B$ be a proper submersion whose fibers are odd dimensional oriented spin manifolds and $E\to X$ a complex vector bundle. After putting the necessary geometric data (metrics, horizontal distribution, various unitary connections), we have a spin Dirac operator $D$. To define the analytic index $\textrm{ind}(D)\in K^{-1}(B)$, we define an operator $D_\theta$ following (3.2) of page 81 of Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry III". Then we get an index $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$. Since its restriction to $\{0\}\times B$ is trivial, it lies in $\ker(i^*:K^0(\mathbb{S}^1\times B)\to K^0(B))\cong K^{-1}(B)$, where $i:B\to\mathcal{S}^1\times B$ is the inclusion map, and $\textrm{ind}(D)\in K^{-1}(B)$ is defined to be $$\textrm{ind}(D)=\textrm{ind}(D_\theta)\in K^{-1}(B).$$ More precisely, the right-hand side of the above should be regarded as the corresponding element.

My questions are

  1. If, by some luck, the family of kernels $\ker(D_b)$ form a vector bundle (which is ungraded), does $\ker(D)\to B$ represent $\textrm{ind}(D)\in K^{-1}(B)$?

  2. If the answer to Q.1 is yes, what can we say about $\ker(D)\to B$ and $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$? Or more precisely, if $\ker(D)\to B$ exists, does $\ker(D_\theta)\to B$ exist? If so, how are they related?

  3. If the answer to Q.1 is no, is the "suspension trick" the only way to define $\textrm{ind}(D)$? It's well known that in the even-dimensional fiber case, if $\ker(D)\to B$ does not exist one can perturb $D$ so that $\textrm{ind}(D)=[\ker(D')]-[\mathbb{C}^n]$, where $D'$ is the perturbed operator and $[\mathbb{C}^n]$ means it is represented by a trivial bundle of rank $n$ for some $n$. Is there any "perturbation method" in the odd dimensional fiber case?

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the underlying setting of my questions is given by the local family index theorem with odd dimensional fibers. My questions are somehow related to the post "https://mathoverflow.net/questions/43175/index-of-a-family-of-dirac-operators-in-k1".

The background of my questions is the following. Let $\pi:X\to B$ be a proper submersion whose fibers are odd dimensional oriented spin manifolds and $E\to X$ a complex vector bundle. After putting the necessary geometric data (metrics, horizontal distribution, various unitary connections), we have a spin Dirac operator $D$. To define the analytic index $\textrm{ind}(D)\in K^{-1}(B)$, we define an operator $D_\theta$ following (3.2) of page 81 of Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry III". Then we get an index $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$. Since its restriction to $\{0\}\times B$ is trivial, it lies in $\ker(i^*:K^0(\mathbb{S}^1\times B)\to K^0(B))\cong K^{-1}(B)$, where $i:B\to\mathcal{S}^1\times B$ is the inclusion map, and $\textrm{ind}(D)\in K^{-1}(B)$ is defined to be $$\textrm{ind}(D)=\textrm{ind}(D_\theta)\in K^{-1}(B).$$ More precisely, the right-hand side of the above should be regarded as the corresponding element.

My questions are

  1. If, by some luck, the family of kernels $\ker(D_b)$ form a vector bundle (which is ungraded), does $\ker(D)\to B$ represent $\textrm{ind}(D)\in K^{-1}(B)$?

  2. If the answer to Q.1 is yes, what can we say about $\ker(D)\to B$ and $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$? Or more precisely, if $\ker(D)\to B$ exists, does $\ker(D_\theta)\to B$ exist? If so, how are they related?

  3. Let's suppose $\ker(D_b)$ do not form a bundle. It's well known that in the even-dimensional fiber case if $\ker(D_\theta)\to B$ does not exist one can perturb $D_\theta$ so that $\textrm{ind}(D_\theta)=[\ker(D_\theta')]-[\mathbb{C}^n]$, where $D_\theta'$ is the perturbed operator and $[\mathbb{C}^n]$ means it is represented by a trivial bundle of rank $n$ for some $n$. Is there any "perturbation method" in the odd dimensional fiber case? So I meant, is there any way to define $\textrm{ind}(D)\in K^{-1}(B)$ directly even though $\ker(D_b)$ do not form a bundle (this kind of implicitly assumes that the answer to Q.1 is yes) without going through the "suspension trick"?

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Ho Man-Ho
Ho Man-Ho
  • 1.2k
  • 6
  • 11

Some questions on defining the analytic index

The questions I have are about the definition of the analytic index of a family of self-adjoint Fredholm operators parameterized by a compact space $B$ (say a closed manifold). Actually, the underlying setting of my questions is given by the local family index theorem with odd dimensional fibers. My questions are somehow related to the post "https://mathoverflow.net/questions/43175/index-of-a-family-of-dirac-operators-in-k1".

The background of my questions is the following. Let $\pi:X\to B$ be a proper submersion whose fibers are odd dimensional oriented spin manifolds and $E\to X$ a complex vector bundle. After putting the necessary geometric data (metrics, horizontal distribution, various unitary connections), we have a spin Dirac operator $D$. To define the analytic index $\textrm{ind}(D)\in K^{-1}(B)$, we define an operator $D_\theta$ following (3.2) of page 81 of Atiyah-Patodi-Singer's paper "Spectral asymmetry and Riemannian geometry III". Then we get an index $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$. Since its restriction to $\{0\}\times B$ is trivial, it lies in $\ker(i^*:K^0(\mathbb{S}^1\times B)\to K^0(B))\cong K^{-1}(B)$, where $i:B\to\mathcal{S}^1\times B$ is the inclusion map, and $\textrm{ind}(D)\in K^{-1}(B)$ is defined to be $$\textrm{ind}(D)=\textrm{ind}(D_\theta)\in K^{-1}(B).$$ More precisely, the right-hand side of the above should be regarded as the corresponding element.

My questions are

  1. If, by some luck, the family of kernels $\ker(D_b)$ form a vector bundle (which is ungraded), does $\ker(D)\to B$ represent $\textrm{ind}(D)\in K^{-1}(B)$?

  2. If the answer to Q.1 is yes, what can we say about $\ker(D)\to B$ and $\textrm{ind}(D_\theta)\in K^0(\mathbb{S}^1\times B)$? Or more precisely, if $\ker(D)\to B$ exists, does $\ker(D_\theta)\to B$ exist? If so, how are they related?

  3. If the answer to Q.1 is no, is the "suspension trick" the only way to define $\textrm{ind}(D)$? It's well known that in the even-dimensional fiber case, if $\ker(D)\to B$ does not exist one can perturb $D$ so that $\textrm{ind}(D)=[\ker(D')]-[\mathbb{C}^n]$, where $D'$ is the perturbed operator and $[\mathbb{C}^n]$ means it is represented by a trivial bundle of rank $n$ for some $n$. Is there any "perturbation method" in the odd dimensional fiber case?