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equivariant cohomotopy

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Homotopy theory

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Idea

The Bredon cohomology-equivariant enhancement of cohomotopy-theory is equivariant cohomotopy:

For GG a group, VV a finite-dimensional real GG-representation GO(V)G \to O(V) and writing S VS^V for the corresponding representation sphere, the equivariant cohomotopy in RO(G)-degree VV of a G-space XX is the set of GG-equivariant homotopy classes of maps from XX to S VS^V:

π G V(X)[X,S V] G. \pi_G^V\big( X \big) \;\coloneqq\; \left[ X, S^V \right]^G \,.

For V= nV = \mathbb{R}^n the trivial representation of dimension nn, this reduces to the definition of plain cohomotopy-sets

π n(X)=π n(X)=[X,S n]. \pi^{\mathbb{R}^n}(X) \;=\; \pi^n(X) \;=\; \left[ X, S^n\right] \,.

The stabilization of this construction, in the sense of equivariant stable homotopy theory, yields the concept of equivariant stable cohomotopy.

Properties

Equivariant Hopf degree theorem

Examples

Equivariant Cohomotopy of S VS^V in RO-degree VV

As a special case of the equivariant Hopf degree theorem , we obtain the following:

Proposition

(equivariant cohomotopy of representation sphere S VS^V in RO(G)-degree VV)

Let GGrp finG \in \mathrm{Grp}_{\mathrm{fin}} and VRO(G)V \in \mathrm{RO}(G) with V G=0V^G = 0. Then the bipointed equivariant cohomotopy of the representation sphere S VS^V in RO(G)-degree VV is the Cartesian product of one copy of the integers for each isotropy subgroup (?) of GG in S VS^V except the full subgroup GGG \subset G

π V(S V) {0,}/ HIsotr S V(G)HG|W G(H)| [S VcS V] (Hdeg(c H)offs(c,H)) \array{ \pi^V\left( S^V\right)^{\{0,\infty\}/} & \overset{\simeq}{\longrightarrow} & \underset{ { { {H \in \mathrm{Isotr}_{S^V}(G)} \atop {H \neq G} } } }{\prod} \;\; {\vert W_G(H)\vert } \cdot \mathbb{Z} \\ \big[ S^V \overset{c}{\longrightarrow} S^V \big] &\mapsto& \Big( H \mapsto \mathrm{deg} \big( c^H \big) - \mathrm{offs}(c,H) \Big) }

where on the right

deg((S V) Hc H(S V) H) \mathrm{deg} \Big( \big( S^V \big)^H \overset{ c^H }{\longrightarrow} \big( S^V \big)^H \Big) \in \mathbb{Z}

is the integer winding number of the underlying continuous function of cc (co)restricted to HH-fixed points, and part of the claim is that this is an integer multiple of the order of the Weyl group W G(H)W_G(H) up to an offset

offs(f,H){0,1,,|W G(H)|} \mathrm{offs}(f,H) \;\in\; \big\{ 0,1, \cdots, \left\vert W_G(H)\right\vert \big\} \;\subset\; \mathbb{Z}

which depends in a definite way on the degrees of c Kc^K for all isotropy groups K>HK \gt H.

For proof see here at equivariant Hopf degree theorem.

Example

(equivariant cohomotopy of S sgnS^{\mathbb{R}_{sgn}} in RO(G)-degree the sign representation sgn\mathbb{R}_{sgn})

Let G= 2G = \mathbb{Z}_2 the cyclic group of order 2 and sgnRO( 2)\mathbb{R}_{sgn} \in RO(\mathbb{Z}_2) its 1-dimensional sign representation.

Under equivariant stereographic projection (here) the corresponding representation sphere S sgnS^{\mathbb{R}_{sgn}} is equivalently the unit circle

S 1S( 2) S^1 \simeq S(\mathbb{R}^2)

equipped with the 2\mathbb{Z}_2-action whose involution element σ\sigma reflects one of the two coordinates of the ambient Cartesian space

σ:(x 1,x 2)(x 1,x 2). \sigma \;\colon\; (x_1,x_2) \mapsto (x_1, -x_2) \,.

Equivalently, if we identify

(1)S 1/ S^1 \;\simeq\; \mathbb{R}/\mathbb{Z}

then the involution action is

σ:t 1t 1t. \begin{aligned} \sigma \;\colon\; t \mapsto & \phantom{\sim} 1 - t \\ & \sim \phantom{1} - t \end{aligned} \,.

This means that the fixed point space is the 0-sphere

(S 1) 2S 0 \big( S^1\big)^{\mathbb{Z}_2} \;\simeq\; S^0

being two antipodal points on the circle, which in the presentation (1) are labeled {0,1/2}S 0\{0,1/2\} \simeq S^0.

Notice that the map

(2)S 1 n S 1 t nt \array{ S^1 &\overset{n}{\longrightarrow}& S^1 \\ t &\mapsto& n\cdot t }

of constant parameter speed and winding number nn \in \mathbb{N} is equivariant for this 2\mathbb{Z}_2-action on both sides:

Now the restriction of the map n()n \cdot (-)\in \mathbb{Z} from (2) to the fixed points

S 0=(S sgn) S sgn (n) 2 n S 0=(S sgn) S sgn \array{ S^0 = \left( S^{\mathbb{R}_{sgn}}\right) &\hookrightarrow& S^{\mathbb{R}_{sgn}} \\ {}^{ \mathllap{ \left( \cdot n\right)^{\mathbb{Z}_2} } } \big\downarrow && \big\downarrow^{\mathrlap{\cdot n}} \\ S^0 = \left( S^{\mathbb{R}_{sgn}}\right) &\hookrightarrow& S^{\mathbb{R}_{sgn}} }

sends (0 to 0 and) 1/21/2 to either 1/21/2 or to 00, depending on whether the winding number is odd or even:

S 0 (n) 2 S 0 1/2 {1/2 | nis odd 0 | nis even \array{ S^0 &\overset{ \left(\cdot n\right)^{\mathbb{Z}_2} }{\longrightarrow}& S^0 \\ 1/2 &\mapsto& \left\{ \array{ 1/2 &\vert& n \;\text{is odd} \\ 0 &\vert& n \text{is even} } \right. }

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 kk \in \mathbb{Z} corresponds to the odd winding integer 1+2k1 + 2k

π sgn(S sgn) {0,}/ 2+1 [/c/] 001/21/2 deg(c) (deg(c)1)/2 \array{ \pi^{\mathbb{R}_{sgn}} \left( S^{\mathbb{R}_{sgn}} \right)^{\{0,\infty\}/} &\simeq& 2 \cdot \mathbb{Z} + 1 &\simeq& \mathbb{Z} \\ \left[ \mathbb{R}/\mathbb{Z} \overset{c}{\to} \mathbb{R}/\mathbb{Z} \right]_{{0 \mapsto 0} \atop {1/2 \mapsto 1/2}} &\mapsto& deg(c) &\mapsto& \big( deg(c) - 1\big)/2 }
Example

(equivariant cohomotopy of S S^{\mathbb{H}} in RO(G)-degree the quaternions \mathbb{H})

Let GSU(2)S()G \subset SU(2) \simeq S(\mathbb{H}) be a non-trivial finite subgroup of SU(2) and let RO(G)\mathbb{H} \in RO(G) be the real vector space of quaternions regarded as a linear representation of GG by left multiplication with unit quaternions.

Then the bi-pointed equivariant cohomotopy of the representation sphere S S^{\mathbb{H}} in RO(G)-degree \mathbb{H} is

π (S ) {0,}/ |G|+1 |G| [S cS ] deg(c {e}) deg(c {e})1 (deg(c {e})1)/|G| \array{ \pi^{\mathbb{H}} \left( S^{\mathbb{H}} \right)^{\{0,\infty\}/} &\simeq& {\left\vert G\right\vert} \cdot \mathbb{Z} + 1 &\simeq& {\left\vert G\right\vert} \cdot \mathbb{Z} &\simeq& \mathbb{Z} \\ \left[ S^{\mathbb{H}} \overset{c}{\longrightarrow} S^{\mathbb{H}} \right] &\mapsto& deg\left( c^{ \{e\} }\right) &\mapsto& deg\left( c^{ \{e\} }\right) - 1 &\mapsto& \big( deg\left( c^{ \{e\} }\right) - 1 \big)/ {\left\vert G\right\vert} }
Proof

The only isotropy subgroups of the left action of GG on \mathbb{H} are the two extreme cases Isotr (G)={1,G}Sub(G)Isotr_{\mathbb{H}}(G) = \{1, G\} \in Sub(G). Hence the only multiplicity that appears in Prop. is

|W G(1)|=|G|. \left\vert W_G(1)\right\vert \;=\; \left\vert G \right\vert \,.

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 offs(id S ,1)=deg(id S )=1offs\left(id_{S^{\mathbb{H}}},1\right) = deg\left( id_{S^{\mathbb{H}}}\right) = 1.

flavours of
Cohomotopy
cohomology theory
cohomology
(full or rational)
equivariant cohomology
(full or rational)
non-abelian cohomologyCohomotopy
(full or rational)
equivariant Cohomotopy
twisted cohomology
(full or rational)
twisted Cohomotopytwisted equivariant Cohomotopy
stable cohomology
(full or rational)
stable Cohomotopyequivariant stable Cohomotopy


References

General

For M-brane charge quantization

Discussion of M-brane charge quantization in equivariant cohomotopy:

Last revised on October 6, 2019 at 10:20:33. See the history of this page for a list of all contributions to it.