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| Format: | Preprint |
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2021
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| Accès en ligne: | https://arxiv.org/abs/2111.08812 |
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| _version_ | 1866914765969817600 |
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| author | Kuhn, Nicholas J. Lloyd, Christopher J. R. |
| author_facet | Kuhn, Nicholas J. Lloyd, Christopher J. R. |
| contents | We study K(n)*(Gr(d,m)) for all n - the 2-local Morava K-theories of the real Grassmanian Gr(d,m) of d-planes in R^m, about which very little has been previously computed. We conjecture that the Atiyah-Hirzebruch Spectral Sequences computing these all collapse after the first possible non-zero differential, and give much evidence that this is the case. Computational patterns for all n seem similar to the known calculation of H*(Gr(d,m);Q), the n=0 case.
We use a novel method to show that higher differentials can't occur: we get a lower bound on the size of K(n)*(Gr(d,m)) by constructing an action of C = the cyclic group of order 4, on our Grassmanians, and then applying the chromatic fixed point theory of the authors' previous paper. In essence, we bound the size of K(n)*(Gr(d,m)) from below by computing K(n-1)*(Gr(d,m)^C).
Meanwhile, the AHSS after the first differential is determined by Q_n-homology, where Q_n is Milnor's nth primitive operation in mod 2 cohomology. Whenever we are able to calculate this, we have found that it agrees with our lower bound for the size of K(n)*(Gr(d,m)). We have two general families where we prove this: m at most 2^{n+1} and all d, and d=2 and all m and n. Computer calculations have allowed us to check many other examples with larger values of d. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_08812 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Computing the Morava K-theory of real Grassmanians using chromatic fixed point theory Kuhn, Nicholas J. Lloyd, Christopher J. R. Algebraic Topology 55M35 We study K(n)*(Gr(d,m)) for all n - the 2-local Morava K-theories of the real Grassmanian Gr(d,m) of d-planes in R^m, about which very little has been previously computed. We conjecture that the Atiyah-Hirzebruch Spectral Sequences computing these all collapse after the first possible non-zero differential, and give much evidence that this is the case. Computational patterns for all n seem similar to the known calculation of H*(Gr(d,m);Q), the n=0 case. We use a novel method to show that higher differentials can't occur: we get a lower bound on the size of K(n)*(Gr(d,m)) by constructing an action of C = the cyclic group of order 4, on our Grassmanians, and then applying the chromatic fixed point theory of the authors' previous paper. In essence, we bound the size of K(n)*(Gr(d,m)) from below by computing K(n-1)*(Gr(d,m)^C). Meanwhile, the AHSS after the first differential is determined by Q_n-homology, where Q_n is Milnor's nth primitive operation in mod 2 cohomology. Whenever we are able to calculate this, we have found that it agrees with our lower bound for the size of K(n)*(Gr(d,m)). We have two general families where we prove this: m at most 2^{n+1} and all d, and d=2 and all m and n. Computer calculations have allowed us to check many other examples with larger values of d. |
| title | Computing the Morava K-theory of real Grassmanians using chromatic fixed point theory |
| topic | Algebraic Topology 55M35 |
| url | https://arxiv.org/abs/2111.08812 |