nLab equivariant cohomotopy

Contents

Context

Representation theory

representation theory

geometric representation theory

Theorems

Spheres

n-sphere

low dimensional n-spheres

Contents

Idea

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

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

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

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

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

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

Examples

Equivariant Cohomotopy of $S^V$ in RO-degree $V$

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

Proposition

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

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

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

where on the right

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

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

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

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

Example

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

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

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

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

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

$\sigma \;\colon\; (x_1,x_2) \mapsto (x_1, -x_2) \,.$

Equivalently, if we identify

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

then the involution action is

\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

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

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

Notice that the map

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

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

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

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

sends (0 to 0 and) $1/2$ to either $1/2$ or to $0$, depending on whether the winding number is odd or 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 $k \in \mathbb{Z}$ corresponds to the odd winding integer $1 + 2k$

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

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

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

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

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

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

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

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

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
differential cohomologydifferential Cohomotopyequivariant differential cohomotopy
persistent cohomologypersistent Cohomotopypersistent equivariant Cohomotopy

References

Cocycle spaces

Discussion of cocycle spaces in equivariant Cohomotopy:

For M-brane charge quantization

Discussion of M-brane charge quantization in equivariant cohomotopy:

Last revised on December 21, 2021 at 12:01:34. See the history of this page for a list of all contributions to it.