Inverse problem : Finding a vector bundle (resp. with connection), given its characteristic class (resp....












1














Given a rank $n$vector bundle $alpha :E to M$, and an element $u in H^k(BG, mathbb{Z})$, $G=GL(n,mathbb{R})$we can define its characteristic class $u(alpha) in H^k(M, mathbb{Z})$ as $f_alpha^* u$ where $f_{alpha}$ is any classifying map.
I am interested in knowing :




(i) What elements $ xin H^k(M, mathbb{Z})$ arise as characteristic classes of some bundle, and some choice of $u in H^k(BG, mathbb{Z})$ ? i.e. What are the necessary and sufficient conditions to be satisfied by $x$ ?



(ii) given an $x in H^k(M, mathbb{Z})$ which is the characteristic class of some bundle, is there a known way to construct the bundle ?




The corresponding questions for vector bundles with connection are




(iii) Given a differential character $ chi in hat{H}(2k)(M)$, what are the conditions to be satisfied by $chi$ so that it is the differential characters of some bundle $E to M$ with connection $theta$ for some choice of the compatible pair $(P,u) in I_{0}^{k}(G) times H^{2k}(BG,mathbb{Z})$ ?



(iv) How does one construct the bundle and the connection, given the data ?




Of course the questions are related to each other. If the characteristic class of a character is not obtainable as the characteristic class of a bundle, clearly the character does not arise from a bundle with connection.
If the answers are not known in full generality, I would still like to know some special cases for which the questions can be answered. One such special case that I know is of course the complex line bundles and their chern classes.
And I would greatly appreciate any references where I may find any clues/answers.

Thanks a lot !






differential-geometry reference-request algebraic-topology vector-bundles characteristic-classes







share|cite|improve this question


















  • 1




    I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
    – Tyrone
    Dec 9 at 12:06










  • @Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
    – user90041
    Dec 9 at 14:01






  • 1




    I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
    – Tyrone
    Dec 9 at 14:35
















1














Given a rank $n$vector bundle $alpha :E to M$, and an element $u in H^k(BG, mathbb{Z})$, $G=GL(n,mathbb{R})$we can define its characteristic class $u(alpha) in H^k(M, mathbb{Z})$ as $f_alpha^* u$ where $f_{alpha}$ is any classifying map.
I am interested in knowing :




(i) What elements $ xin H^k(M, mathbb{Z})$ arise as characteristic classes of some bundle, and some choice of $u in H^k(BG, mathbb{Z})$ ? i.e. What are the necessary and sufficient conditions to be satisfied by $x$ ?



(ii) given an $x in H^k(M, mathbb{Z})$ which is the characteristic class of some bundle, is there a known way to construct the bundle ?




The corresponding questions for vector bundles with connection are




(iii) Given a differential character $ chi in hat{H}(2k)(M)$, what are the conditions to be satisfied by $chi$ so that it is the differential characters of some bundle $E to M$ with connection $theta$ for some choice of the compatible pair $(P,u) in I_{0}^{k}(G) times H^{2k}(BG,mathbb{Z})$ ?



(iv) How does one construct the bundle and the connection, given the data ?




Of course the questions are related to each other. If the characteristic class of a character is not obtainable as the characteristic class of a bundle, clearly the character does not arise from a bundle with connection.
If the answers are not known in full generality, I would still like to know some special cases for which the questions can be answered. One such special case that I know is of course the complex line bundles and their chern classes.
And I would greatly appreciate any references where I may find any clues/answers.

Thanks a lot !






differential-geometry reference-request algebraic-topology vector-bundles characteristic-classes







share|cite|improve this question


















  • 1




    I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
    – Tyrone
    Dec 9 at 12:06










  • @Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
    – user90041
    Dec 9 at 14:01






  • 1




    I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
    – Tyrone
    Dec 9 at 14:35














1












1








1


1





Given a rank $n$vector bundle $alpha :E to M$, and an element $u in H^k(BG, mathbb{Z})$, $G=GL(n,mathbb{R})$we can define its characteristic class $u(alpha) in H^k(M, mathbb{Z})$ as $f_alpha^* u$ where $f_{alpha}$ is any classifying map.
I am interested in knowing :




(i) What elements $ xin H^k(M, mathbb{Z})$ arise as characteristic classes of some bundle, and some choice of $u in H^k(BG, mathbb{Z})$ ? i.e. What are the necessary and sufficient conditions to be satisfied by $x$ ?



(ii) given an $x in H^k(M, mathbb{Z})$ which is the characteristic class of some bundle, is there a known way to construct the bundle ?




The corresponding questions for vector bundles with connection are




(iii) Given a differential character $ chi in hat{H}(2k)(M)$, what are the conditions to be satisfied by $chi$ so that it is the differential characters of some bundle $E to M$ with connection $theta$ for some choice of the compatible pair $(P,u) in I_{0}^{k}(G) times H^{2k}(BG,mathbb{Z})$ ?



(iv) How does one construct the bundle and the connection, given the data ?




Of course the questions are related to each other. If the characteristic class of a character is not obtainable as the characteristic class of a bundle, clearly the character does not arise from a bundle with connection.
If the answers are not known in full generality, I would still like to know some special cases for which the questions can be answered. One such special case that I know is of course the complex line bundles and their chern classes.
And I would greatly appreciate any references where I may find any clues/answers.

Thanks a lot !






differential-geometry reference-request algebraic-topology vector-bundles characteristic-classes







share|cite|improve this question













Given a rank $n$vector bundle $alpha :E to M$, and an element $u in H^k(BG, mathbb{Z})$, $G=GL(n,mathbb{R})$we can define its characteristic class $u(alpha) in H^k(M, mathbb{Z})$ as $f_alpha^* u$ where $f_{alpha}$ is any classifying map.
I am interested in knowing :




(i) What elements $ xin H^k(M, mathbb{Z})$ arise as characteristic classes of some bundle, and some choice of $u in H^k(BG, mathbb{Z})$ ? i.e. What are the necessary and sufficient conditions to be satisfied by $x$ ?



(ii) given an $x in H^k(M, mathbb{Z})$ which is the characteristic class of some bundle, is there a known way to construct the bundle ?




The corresponding questions for vector bundles with connection are




(iii) Given a differential character $ chi in hat{H}(2k)(M)$, what are the conditions to be satisfied by $chi$ so that it is the differential characters of some bundle $E to M$ with connection $theta$ for some choice of the compatible pair $(P,u) in I_{0}^{k}(G) times H^{2k}(BG,mathbb{Z})$ ?



(iv) How does one construct the bundle and the connection, given the data ?




Of course the questions are related to each other. If the characteristic class of a character is not obtainable as the characteristic class of a bundle, clearly the character does not arise from a bundle with connection.
If the answers are not known in full generality, I would still like to know some special cases for which the questions can be answered. One such special case that I know is of course the complex line bundles and their chern classes.
And I would greatly appreciate any references where I may find any clues/answers.

Thanks a lot !






differential-geometry reference-request algebraic-topology vector-bundles characteristic-classes




differential-geometry reference-request algebraic-topology vector-bundles characteristic-classes






share|cite|improve this question













share|cite|improve this question











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asked Dec 8 at 23:48









user90041

1,7541233




1,7541233








  • 1




    I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
    – Tyrone
    Dec 9 at 12:06










  • @Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
    – user90041
    Dec 9 at 14:01






  • 1




    I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
    – Tyrone
    Dec 9 at 14:35














  • 1




    I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
    – Tyrone
    Dec 9 at 12:06










  • @Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
    – user90041
    Dec 9 at 14:01






  • 1




    I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
    – Tyrone
    Dec 9 at 14:35








1




1




I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
– Tyrone
Dec 9 at 12:06




I think you need more conditions on $M$ and $G$ to be able to say anything worthwhile. You are essentially asking for a classification of homotopy classes of maps $Mrightarrow BG$, and this a difficult problem in general. Even if you take $M=S^n$, then the question more or less boils down to which elements of $H_*BG$ are in the image of the Hurewicz map and even this is not known. As for $(ii)$, just choose a classifying map $f$ such that $f^*u=x$ and take $E=f^*EG$ for an explicit construction.
– Tyrone
Dec 9 at 12:06












@Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
– user90041
Dec 9 at 14:01




@Tyrone Thanks for the comment.You are right, the question is very broad. Can one say something about the following special case : Suppose $chi$ is a differential character whose characteristic class $u(chi)$, and curvature $P(chi)$ both vanish. Is it possible to say whether this character arises from a vector bundle with connection ?
– user90041
Dec 9 at 14:01




1




1




I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
– Tyrone
Dec 9 at 14:35




I'm afraid that I don't know any of the theory behind differential characters, so I don't feel too confident answering your second set of question. I'm interested to hear any answers, though, so I upvoted your question. I would suggest maybe limiting yourself to the classical groups $O(n)$ or $U(n)$ and maybe to restrict to manifolds whose dimensions fall in the respective stable ranges.
– Tyrone
Dec 9 at 14:35















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