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DTSTART:20230312T070000
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URL:https://uwaterloo.ca/pure-mathematics-geometry-topology/events/topologi
 cal-aspects-almost-complex-structures-six-sphere
SUMMARY:Topological aspects of almost complex structures on the six sphere
CLASS:PUBLIC
DESCRIPTION:ALEKSANDAR MILIVOJEVIC\, MAX PLANCK INSTITUTE FOR MATHEMATICS\,
  BONN\n\nBy thinking of the six-sphere S6 as the unit sphere in the imagi
 nary\noctonions\, one detects a real projective seven-space RP7 in the sp
 ace\nof all almost complex structures on S6. On the other hand\, using the
 \nHaefliger-Sullivan rational homotopy theoretic model for the space of\ns
 ections of a fiber bundle applied to the twistor space construction\,\none
  can abstractly calculate that the rational homology of the space\nof (ori
 entation-compatible) almost complex structures on S6 agrees\nwith that of
  RP7. Sullivan asked whether the inclusion of the\noctonionic RP7 into th
 e space of all almost complex structures is a\nhomotopy equivalence. We sh
 ow that it is not\, though it is a rational\nhomology equivalence that ind
 uces an isomorphism on fundamental\ngroups. We can further describe the ho
 motopy fiber of this\ninclusion. \n\nOn a related note\, over six-manifol
 ds\, almost complex structures\ncorrespond to embedded half-dimensional su
 bmanifolds of the twistor\nspace\, and hence one obtains numerical invaria
 nts via their\nhomological intersection. Time permitting\, we compute thes
 e numbers\nconcretely over the six-sphere and other six-manifolds\, and co
 mment on\ntheir relation to integrability. \n\nThis is joint work with Bo
 ra Ferlengez and Gustavo Granja.\n\nMC 5417
DTSTAMP:20260311T030316Z
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