What are interesting examples of still uncomputed cohomology rings?

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It seems, that very many cohomology rings of interesting spaces have been computed. For example the integral cohomology rings of $SO(n)$, $Spin(n)$, $U(n)$, $SU(n)$, $Sp(n)$, the real and complex Grassmannians, the real and complex Stiefel manifolds are known. What are the most important examples of spaces of which the (singular) cohomology (with some coeffitients) is not known?




algebraic-topology soft-question homology-cohomology




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  • Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
    – freakish
    Jul 23 at 9:39







  • 1




    Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
    – Takirion
    Jul 23 at 10:21






  • 1




    @freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
    – Najib Idrissi
    Jul 23 at 11:04







  • 1




    @freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
    – Najib Idrissi
    Jul 23 at 12:26







  • 1




    So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
    – Takirion
    Jul 23 at 16:20














up vote
10
down vote

favorite
3












It seems, that very many cohomology rings of interesting spaces have been computed. For example the integral cohomology rings of $SO(n)$, $Spin(n)$, $U(n)$, $SU(n)$, $Sp(n)$, the real and complex Grassmannians, the real and complex Stiefel manifolds are known. What are the most important examples of spaces of which the (singular) cohomology (with some coeffitients) is not known?




algebraic-topology soft-question homology-cohomology




share|cite|improve this question





















  • Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
    – freakish
    Jul 23 at 9:39







  • 1




    Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
    – Takirion
    Jul 23 at 10:21






  • 1




    @freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
    – Najib Idrissi
    Jul 23 at 11:04







  • 1




    @freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
    – Najib Idrissi
    Jul 23 at 12:26







  • 1




    So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
    – Takirion
    Jul 23 at 16:20












up vote
10
down vote

favorite
3









up vote
10
down vote

favorite
3






3





It seems, that very many cohomology rings of interesting spaces have been computed. For example the integral cohomology rings of $SO(n)$, $Spin(n)$, $U(n)$, $SU(n)$, $Sp(n)$, the real and complex Grassmannians, the real and complex Stiefel manifolds are known. What are the most important examples of spaces of which the (singular) cohomology (with some coeffitients) is not known?




algebraic-topology soft-question homology-cohomology




share|cite|improve this question













It seems, that very many cohomology rings of interesting spaces have been computed. For example the integral cohomology rings of $SO(n)$, $Spin(n)$, $U(n)$, $SU(n)$, $Sp(n)$, the real and complex Grassmannians, the real and complex Stiefel manifolds are known. What are the most important examples of spaces of which the (singular) cohomology (with some coeffitients) is not known?





algebraic-topology soft-question homology-cohomology





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 23 at 6:34
























asked Jul 22 at 10:20









Takirion

581211




581211











  • Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
    – freakish
    Jul 23 at 9:39







  • 1




    Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
    – Takirion
    Jul 23 at 10:21






  • 1




    @freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
    – Najib Idrissi
    Jul 23 at 11:04







  • 1




    @freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
    – Najib Idrissi
    Jul 23 at 12:26







  • 1




    So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
    – Takirion
    Jul 23 at 16:20
















  • Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
    – freakish
    Jul 23 at 9:39







  • 1




    Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
    – Takirion
    Jul 23 at 10:21






  • 1




    @freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
    – Najib Idrissi
    Jul 23 at 11:04







  • 1




    @freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
    – Najib Idrissi
    Jul 23 at 12:26







  • 1




    So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
    – Takirion
    Jul 23 at 16:20















Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
– freakish
Jul 23 at 9:39





Every space that is homotopy equivalent to a CW complex (and this covers lots of spaces, in particular all manifolds) has computable (co)homology as long as you know the CW decomposition. So I think that if such examples exist they are most likely "pathological".
– freakish
Jul 23 at 9:39





1




1




Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
– Takirion
Jul 23 at 10:21




Yes, but one of the points is, that one does not always know a CW decomposition. Consider for example an Eilenberg-MacLane Space $K(G,n)$ for some group $G$ (it doesn't even have to be very complicated). Then it totally unclear from the definition of $K(G,n)$ what a CW-decomposition could be, as one has to deal with (complicated) homotopy groups. And even if you have got a CW-decomposition you would still have to analyse it. So one could rephrase the questions as: Are there interesing examples, where the CW-structure has not been found, or analysed?
– Takirion
Jul 23 at 10:21




1




1




@freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
– Najib Idrissi
Jul 23 at 11:04





@freakish That's a bit optimistic. It would be much more complicated to "write down" a CW decomposition (whatever that means) of a random space than it would be to compute its cohomology. For example, I dare you to try and write down a CW decomposition of $GL_n(mathbbR)$ – and yet computing its cohomology is very easy... It's a bit like saying that computing with groups is easy, because every group has a presentation. But then how do you find the presentation, and how do you make actual computations with that presentation that may be horribly unwieldy?
– Najib Idrissi
Jul 23 at 11:04





1




1




@freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
– Najib Idrissi
Jul 23 at 12:26





@freakish Really, the question is "are there spaces for which the cohomology of configuration spaces is known?" Until recently, to my knowledge, it was not possible to compute the cohomology ring of configuration spaces of manifolds in full generality. Since then there has been some results in my own work (partly j/w Campos, Lambrechts, Willwacher). If you are just interested in Betti numbers then you have this whole stability business (too many people to list in this comment). Recently there was a paper of Petersen to compute Betti numbers of configuration spaces of arbitrary spaces.
– Najib Idrissi
Jul 23 at 12:26





1




1




So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
– Takirion
Jul 23 at 16:20




So it seems, that configuration spaces are an important class of spaces with unknown cohomology. Could you maybe explain why these (the spaces and their cohomology) are of special interest or importance? I would really appreciate this as an answer
– Takirion
Jul 23 at 16:20















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