Hopf degree theorem (changes) in nLab

Context

Cohomology

Algebraic topology

• Statement
  • In ordinary homotopy theory
  • In equivariant homotopy theory
• Applications
• Related statements
• References

Definition

(matching pair of GG-spaces)

For GG a finite group, we say that a pair X,Y∈GSpacesX,Y \in G Spaces of topological G-spaces, where XX is a G-CW-complex, is a matching pair if the following conditions are satisfied for all isotropy groups H∈Isotr X(G)H \in Isotr_X(G) (2) of the GG-action on XX.

1. The fixed point space X HX^H is a W GH=(N GH)/HW_G H = (N_G H) / H -complex of finite dimension dim(X H)∈ℕdim\big( X^H\big) \in \mathbb{N};

2. H dim(X H)(X H,ℤ)≃ℤH^{dim(X^H)}\Big( X^H , \mathbb{Z}\Big) \simeq \mathbb{Z} (integral cohomology of the fixed point space),

   this implies that the action of W G(H)W_G(H) on cohomology induces a group homomorphism

   (3)e H,X:W G(H)⟶Aut Ab(ℤ)≃ℤ × e_{H,X} \;\colon\; W_G(H) \longrightarrow Aut_{Ab}(\mathbb{Z}) \simeq \mathbb{Z}^\times

   to be called the orientation behaviour of the action of W G(H)W_G(H) on X HX^H;

and

1. Y HY^H is (dim(X H)−1)(dim(X^H)-1)-connected

   pi <dim(X H)(Y H)=0 pi_{ \lt \mathrm{dim}\big( X^H \big) } \big( Y^H \big) \;=\; 0

   (hence connected if dim(X H)=1dim\left(X^H\right) = 1, simply connected if dim(X H)=2dim\left(X^H\right) = 2, etc.);

2. π dim(X H)(Y H)≃ℤ\pi_{dim(X^H)}\big( Y^H\big) \simeq \mathbb{Z} (homotopy groups of fixed point space),

   with the previous point this implies (by the Hurewicz theorem) that H dim(X H)(Y H,ℤ)≃ℤH^{dim(X^H)}\Big( Y^H , \mathbb{Z}\Big) \simeq \mathbb{Z} and hence orientation behaviour (3) e H,Y:W G(H)→ℤ ×e_{H,Y} \;\colon\; W_G(H) \to \mathbb{Z}^\times

3. e H,X=e H,Ye_{H,X} = e_{H,Y}, the orientation behaviour (3) of XX and YY agrees at all isotropy groups.

For simplicity we also demand that

• dim(X H)≥1dim(X^H) \geq 1.

Given a matching pair of GG-spaces, we say that a choice of generators in the cohomology groups of all the fixed strata

(4)or X,H =±1∈ℤ≃H dim(X H)(X H,ℤ) or Y,H =±1∈ℤ≃H dim(X H)(Y H,ℤ) \begin{aligned} or_{X,H} & = \pm 1 \in \mathbb{Z} \simeq H^{dim(X^H)}\big(X^H, \mathbb{Z} \big) \\ or_{Y,H} & = \pm 1 \in \mathbb{Z} \simeq H^{dim(X^H)}\big(Y^H, \mathbb{Z} \big) \end{aligned}

is a choice of singularity-wise orientations. Given such a choice and an equivariant continuous function f:X→Yf \colon X \to Y, we have for each isotropy group H∈Isotr X(G)H \in Isotr_X(G) that the continuous function f H:X H→Y Hf^H \;\colon\; X^H \to Y^H has a well-defined integer degree

(5)deg(f H)∈ℤ. deg(f^H) \; \in \;\mathbb{Z} \,.

Proof

First notice that an HH-fixed locus of any GG-representation sphere is itself a W G(H)W_G(H)-representation sphere, since (S V) H≃S (V H)\left(S^V\right)^H \simeq S^{\left( V^H\right)}. That these are all WH-CW-complexes follows because generally G-representation spheres are G-CW-complexes. Moreover, since the topological spaces underlying all these fixed loci are n-spheres, and since we have the same spheres for XX and YY, the connectivity and orientability conditions in Def. 1 are evidently satisfied.

Theorem

(equivariant Hopf degree theorem)

Given a matching pair of GG-spaces X,YX, Y (Def. 1) the function (from GG-equivariant homotopy classes to tuples of degrees labeled by isotropy groups) which sends any equivariant homotopy class [f][f] of an equivariant continuou sfunction f:X→Yf \colon X \to Y to its HH-degrees (5)

deg : π 0Maps(X,Y) G ↪AAAA ∏H∈Isotr X(G)ℤ [f] ↦ (H↦deg(f H)) \array{ deg &\colon& \pi_0 \mathrm{Maps}\big( X,Y \big)^G & \overset{ \phantom{AAAA} }{\hookrightarrow} & \underset{ { H \in Isotr_X(G) } }{\prod} \mathbb{Z} \\ && [f] &\mapsto & \big( H \mapsto \mathrm{deg}\big( f^H\big) \big) }

is an injection.

Moreover, for each [f]∈π 0Maps(X,Y) G[f] \in \pi_0 \mathrm{Maps}\big( X,Y \big)^G and for each H∈Isotropy X(G)H \in Isotropy_X(G)

1. the HH-degree (5) modulo the order of the Weyl group

   (6)deg(f H)mod|W G(H)|∈ℤ/|W G(H)| deg\left( f^H\right) \;mod\; {\vert W_G(H)\vert} \;\in\; \mathbb{Z}/{\vert W_G(H)\vert}

   is fixed by the degrees deg(f K)deg\big( f^K\big) for all K>HK \gt H;

2. there exists f′f' with specified degrees deg((f′) K)=deg(f K)deg\big( (f')^K\big) = deg\big( f^K\big) for K>HK \gt H and realizing any of the degrees deg((f′) H)∈ℤdeg\big( (f')^H\big) \in \mathbb{Z} allowed by the constraint (6).

Proposition

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

Let G∈Grp finG \in \mathrm{Grp}_{\mathrm{fin}} and V∈RO(G)V \in \mathrm{RO}(G). Then the pointed 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 proper isotropy subgroup (2) H⊂≠GH \underset{\neq}{\subset} G in S VS^V, and a copy of ℤ 2\mathbb{Z}_2 or ℤ\mathbb{Z} depending on whether V G=0V^G = 0 or not:

π V(S V) {∞}/ ⟶≃ {ℤ 2 | V G=0 ℤ | otherwise}×∏H∈Isotr S V(G)H≠G|W G(H)|⋅ℤ [S V⟶cS V] ↦ (H↦deg(c H)−offs(c,H)) \array{ \pi^V\left( S^V\right)^{\{\infty\}/} & \overset{\simeq}{\longrightarrow} & \left\{ \array{ \mathbb{Z}_2 &\vert& V^G = 0 \\ \mathbb{Z} &\vert& \text{otherwise} } \right\} \times \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) H⟶c H(S V) H)∈{ℤ | dim(V H)>0 ℤ 2 | dim(V H)=0} \mathrm{deg} \Big( \big( S^V \big)^H \overset{ c^H }{\longrightarrow} \big( S^V \big)^H \Big) \in \left\{ \array{ \mathbb{Z} &\vert& dim\left(V^H\right) \gt 0 \\ \mathbb{Z}_2 &\vert& dim\left( V^H\right) = 0 } \right\}

is the winding number of the underlying continuous function of cc (co)restricted to HH-fixed points, and part of the claim is that in the cases with dim(V H)>0dim\left( V^H\right) \gt 0 this is an integer multiple of the order of the Weyl group W G(H)W_G(H) (1) up to an offset

offs(f,H)∈{0,1,⋯,|W G(H)|}⊂ℤ \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.

Proof

This follows as a special case of the equivariant Hopf degree theorem (Theorem 1).

Here (S V,S V)(S^V, S^V) is a matching pair of GG-spaces according to Example 1.

This equivariant Hopf degree theorem is stated above under the simplifying assumption that the dimension of all fixed loci is positive. But the proof from tomDieck 79, 8.4 immediately applies to our situation where the dimension of the fixed locus at the full subgroup H=GH = G may be 0, with (S V) G=S 0\left( S^V\right)^G = S^0. This gives a choice in ℤ 2\mathbb{Z}_2 in the first step of the inductive argument in tomDieck 79, 8.4, and from there on the proof applies verbatim.

Alternatively, if V G=0V^G = 0 we may consider maps S 1+V→S 1+VS^{1+V} \to S^{1+V} which restrict on S 1→S 1S^1 \to S^1 to degree zero or one.

Example

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

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

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

S 1≃S(ℝ 2) S^1 \simeq S(\mathbb{R}^2)

equipped with the ℤ 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

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

then the involution action is

σ:t↦ ∼1−t ∼1−t. \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) ℤ 2≃S 0 \big( S^1\big)^{\mathbb{Z}_2} \;\simeq\; S^0

being two antipodal points on the circle, which in the presentation (7) are labeled {0,1/2}≃S 0\{0,1/2\} \simeq S^0.

Notice that the map

(8)S 1 ⟶n S 1 t ↦ n⋅t \array{ S^1 &\overset{n}{\longrightarrow}& S^1 \\ t &\mapsto& n\cdot t }

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

\begin{center} \begin{xymatrix} t \ar@{|->}[r] \ar@{|->}[d]{\sigma} & n\cdot t \ar@{|->}[d]^{\sigma} \ -t \ar@{|->}[r] & -n \cdot t \end{xymatrix} \end{center}

Now the restriction of the map n⋅(−)∈ℤ n \cdot(-)\in \mathbb{Z} from (8) to the fixed points

S 0=(S ℝ sgn) ↪ S ℝ sgn (⋅n) ℤ 2↓ ↓ ⋅n S 0=(S ℝ sgn) ↪ S ℝ 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) ℤ 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 \;\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. 2 there remains the integers worth of equivariant homotopy classes, where each integer k∈ℤk \in \mathbb{Z} corresponds to the odd winding integer 1+2k1 + 2k

π ℝ sgn(S ℝ sgn) {0,∞}/ ≃ 2⋅ℤ+1 ≃ ℤ [ℝ/ℤ→cℝ/ℤ] 0↦01/2↦1/2 ↦ 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 ℍS^{\mathbb{H}} in RO(G)-degree the quaternions ℍ\mathbb{H})

Let G⊂SU(2)≃S(ℍ)G \subset SU(2) \simeq S(\mathbb{H}) be a non-trivial finite subgroup of SU(2) and let ℍ∈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 ℍS^{\mathbb{H}} in RO(G)-degree ℍ\mathbb{H} is

π ℍ(S ℍ) {0,∞}/ ≃ |G|⋅ℤ+1 ≃ |G|⋅ℤ ≃ ℤ [S ℍ⟶cS ℍ] ↦ 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 ℍ\mathbb{H} are the two extreme cases Isotr ℍ(G)={1,G}∈Sub(G)Isotr_{\mathbb{H}}(G) = \{1, G\} \in Sub(G). Hence the only multiplicity that appears in Prop. 2 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 ℍ,1)=deg(id S ℍ)=1offs\left(id_{S^{\mathbb{H}}},1\right) = deg\left( id_{S^{\mathbb{H}}}\right) = 1.