nLab equivariant cohomotopy

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.

Properties

Equivariant Hopf degree theorem

• equivariant Hopf degree theorem

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 $K \gt H$.

For proof see here at equivariant Hopf degree theorem.

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

Related concepts

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

• equivariant Hopf degree theorem

• equivariant Pontrjagin theorem

References

General

• 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)

Cocycle spaces

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)

For M-brane charge quantization

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]

  (equivariant rational Cohomotopy)

• 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]