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I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same. I am not sure if you meant something more refined like the signature operator.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same. My guess is that you meant something more refined like the signature operator or the operator associated with a spin bundle.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same. I am not sure if you meant something more refined like the signature operator.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I suspect some clarification is needed.

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.