Infinite loop spaces and nilpotent K-theory
Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU, BU, BSO, BO, BSp, BGL∞(R) + and Q0(S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E∞-ring sp...
| Main Authors: | , , , |
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| Format: | Journal article |
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Mathematical Sciences Publishers
2017
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| _version_ | 1826258107912683520 |
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| author | Adem, A Gomez, J Lind, J Tillmann, U |
| author_facet | Adem, A Gomez, J Lind, J Tillmann, U |
| author_sort | Adem, A |
| collection | OXFORD |
| description | Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU, BU, BSO, BO, BSp, BGL∞(R) + and Q0(S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E∞-ring spectra. We introduce the notion of q-nilpotent K-theory of a CW-complex X for any q ≥ 2, which extends the notion of commutative K-theory defined by Adem-G´omez, and show that it is represented by Z × B(q, U), were B(q, U) is the q-th term of the aforementioned filtration of BU. For the proof we introduce an alternative way of associating an infinite loop space to a commutative I-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative I-rig and show that they give rise to non-unital E∞-ring spectra |
| first_indexed | 2024-03-06T18:28:48Z |
| format | Journal article |
| id | oxford-uuid:08e99695-9e2e-4fca-9f42-8cdb615b2864 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:28:48Z |
| publishDate | 2017 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:08e99695-9e2e-4fca-9f42-8cdb615b28642022-03-26T09:15:29ZInfinite loop spaces and nilpotent K-theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:08e99695-9e2e-4fca-9f42-8cdb615b2864Symplectic Elements at OxfordMathematical Sciences Publishers2017Adem, AGomez, JLind, JTillmann, UUsing a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU, BU, BSO, BO, BSp, BGL∞(R) + and Q0(S 0 ). We show that these infinite loop spaces are the zero spaces of non-unital E∞-ring spectra. We introduce the notion of q-nilpotent K-theory of a CW-complex X for any q ≥ 2, which extends the notion of commutative K-theory defined by Adem-G´omez, and show that it is represented by Z × B(q, U), were B(q, U) is the q-th term of the aforementioned filtration of BU. For the proof we introduce an alternative way of associating an infinite loop space to a commutative I-monoid and give criteria when it can be identified with the plus construction on the associated limit space. Furthermore, we introduce the notion of a commutative I-rig and show that they give rise to non-unital E∞-ring spectra |
| spellingShingle | Adem, A Gomez, J Lind, J Tillmann, U Infinite loop spaces and nilpotent K-theory |
| title | Infinite loop spaces and nilpotent K-theory |
| title_full | Infinite loop spaces and nilpotent K-theory |
| title_fullStr | Infinite loop spaces and nilpotent K-theory |
| title_full_unstemmed | Infinite loop spaces and nilpotent K-theory |
| title_short | Infinite loop spaces and nilpotent K-theory |
| title_sort | infinite loop spaces and nilpotent k theory |
| work_keys_str_mv | AT adema infiniteloopspacesandnilpotentktheory AT gomezj infiniteloopspacesandnilpotentktheory AT lindj infiniteloopspacesandnilpotentktheory AT tillmannu infiniteloopspacesandnilpotentktheory |