Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group
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Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group. / Foley, John D.
I: Mathematische Zeitschrift, Bind 301, Nr. 3, 2022, s. 2465-2496.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Recognizing nullhomotopic maps into the classifying space of a Kac–Moody group
AU - Foley, John D.
N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022
Y1 - 2022
N2 - This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac–Moody group and source the classifying space of either a p-compact group or a connected Kac–Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac–Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac–Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations. These partial results show that null maps are detected by restricting to the maximal torus in many cases of interest.
AB - This paper extends certain characterizations of nullhomotopic maps between p-compact groups to maps with target the p-completed classifying space of a connected Kac–Moody group and source the classifying space of either a p-compact group or a connected Kac–Moody group. A well known inductive principle for p-compact groups is applied to obtain general, mapping space level results. An arithmetic fiber square computation shows that a null map from the classifying space of a connected compact Lie group to the classifying space of a connected topological Kac–Moody group can be detected by restricting to the maximal torus. Null maps between the classifying spaces of connected topological Kac–Moody groups cannot, in general, be detected by restricting to the maximal torus due to the nonvanishing of an explicit abelian group of obstructions described here. Nevertheless, partial results are obtained via the application of algebraic discrete Morse theory to higher derived limit calculations. These partial results show that null maps are detected by restricting to the maximal torus in many cases of interest.
KW - Algebraic discrete Morse theory
KW - Arithmetic fiber square
KW - Homotopical group theory
KW - Invariant theory
KW - Kac–Moody groups
KW - p-Compact groups
UR - http://www.scopus.com/inward/record.url?scp=85125035599&partnerID=8YFLogxK
U2 - 10.1007/s00209-022-02981-1
DO - 10.1007/s00209-022-02981-1
M3 - Journal article
AN - SCOPUS:85125035599
VL - 301
SP - 2465
EP - 2496
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 0025-5874
IS - 3
ER -
ID: 310972629