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The Bredon cohomology-equivariant enhancement of cohomotopy-theory is equivariant cohomotopy:
For a group, a finite-dimensional real -representation and writing for the corresponding representation sphere, the equivariant cohomotopy in RO(G)-degree of a G-space is the set of -equivariant homotopy classes of maps from to :
For the trivial representation of dimension , this reduces to the definition of plain cohomotopy-sets
The stabilization of this construction, in the sense of equivariant stable homotopy theory, yields the concept of equivariant stable cohomotopy.
As a special case of the equivariant Hopf degree theorem , we obtain the following:
(equivariant cohomotopy of representation sphere in RO(G)-degree )
Let and with . Then the bipointed equivariant cohomotopy of the representation sphere in RO(G)-degree is the Cartesian product of one copy of the integers for each isotropy subgroup (?) of in except the full subgroup
where on the right
is the integer winding number of the underlying continuous function of (co)restricted to -fixed points, and part of the claim is that this is an integer multiple of the order of the Weyl group up to an offset
which depends in a definite way on the degrees of for all isotropy groups .
For proof see here at equivariant Hopf degree theorem.
(equivariant cohomotopy of in RO(G)-degree the sign representation )
Let the cyclic group of order 2 and its 1-dimensional sign representation.
Under equivariant stereographic projection (here) the corresponding representation sphere is equivalently the unit circle
equipped with the -action whose involution element reflects one of the two coordinates of the ambient Cartesian space
Equivalently, if we identify
then the involution action is
This means that the fixed point space is the 0-sphere
being two antipodal points on the circle, which in the presentation (1) are labeled .
Notice that the map
of constant parameter speed and winding number is equivariant for this -action on both sides:
Now the restriction of the map from (2) to the fixed points
sends (0 to 0 and) to either or to , depending on whether the winding number is odd or even:
Hence if the restriction to the fixed locus is taken to be the identity (bipointed equivariant cohomotopy) then, in accord with Prop. there remains the integers worth of equivariant homotopy classes, where each integer corresponds to the odd winding integer
(equivariant cohomotopy of in RO(G)-degree the quaternions )
Let be a non-trivial finite subgroup of SU(2) and let be the real vector space of quaternions regarded as a linear representation of by left multiplication with unit quaternions.
Then the bi-pointed equivariant cohomotopy of the representation sphere in RO(G)-degree is
The only isotropy subgroups of the left action of on are the two extreme cases . Hence the only multiplicity that appears in Prop. is
and all degrees must differ from that of the class of the identity function by a multiple of this multiplicity. Finally, the offset of the identity function is clearly .
flavours of Cohomotopy cohomology theory | cohomology (full or rational) | equivariant cohomology (full or rational) |
---|---|---|
non-abelian cohomology | Cohomotopy (full or rational) | equivariant Cohomotopy |
twisted cohomology (full or rational) | twisted Cohomotopy | twisted equivariant Cohomotopy |
stable cohomology (full or rational) | stable Cohomotopy | equivariant stable Cohomotopy |
differential cohomology | differential Cohomotopy | equivariant differential cohomotopy |
persistent cohomology | persistent Cohomotopy | persistent equivariant Cohomotopy |
Arthur Wasserman, section 3 of Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)
Tammo tom Dieck, section 8.4 of Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766 Springer 1979
George Peschke, Degree of certain equivariant maps into a representation sphere, Topology and its Applications Volume 59, Issue 2, 30 September 1994, Pages 137-156 (doi:10.1016/0166-8641(94)90091-4)
Zalman Balanov, Equivariant hopf theorem, Nonlinear Analysis: Theory, Methods & Applications Volume 30, Issue 6, December 1997, Pages 3463-3474 (doi:10.1016/S0362-546X(97)00020-5)
James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)
Davide L. Ferrario, On the equivariant Hopf theorem, Topology Volume 42, Issue 2, March 2003, Pages 447-465 (doi:10.1016/S0040-9383(02)00015-0)
Discussion of cocycle spaces in equivariant Cohomotopy:
Victor Vassiliev, Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials (arXiv:1809.05632)
Victor Vassiliev, Cohomology of spaces of Hopf equivariant maps of spheres (arXiv:2102.07157)
Discussion of M-brane charge quantization in equivariant cohomotopy:
John Huerta, Hisham Sati, Urs Schreiber, Real ADE-equivariant (co)homotopy and Super M-branes, Comm. Math. Phys. 371 (2019) 425-524 [arXiv:1805.05987, doi:10.1007/s00220-019-03442-3]
Hisham Sati, Urs Schreiber, Equivariant Cohomotopy implies orientifold tadpole cancellation (arXiv:1909.12277)
(implying RR-field tadpole cancellation)
On quasi-elliptic cohomology of representation spheres as an approximation to equivariant Cohomotopy:
Zhen Huan: Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology [arXiv:2006.00554]
Zhen Huan: Quasi-elliptic cohomology of 4-spheres [arXiv:2408.02278]
Last revised on August 8, 2024 at 05:32:57. See the history of this page for a list of all contributions to it.