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Hopf degree theorem

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Statement

In ordinary homotopy theory

Proposition

(Hopf degree theorem)

Let nn \in \mathbb{N} be a natural number and XMfdX \in Mfd be a connected orientable closed manifold of dimension nn. Then the nnth cohomotopy classes [XcS n]π n(X)\left[X \overset{c}{\to} S^n\right] \in \pi^n(X) of XX are in bijection with the degree deg(c)deg(c) \in \mathbb{Z} of the representing functions, hence the canonical function

π n(X)S nK(,n)H n(X,) \pi^n(X) \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow} H^n(X,\mathbb{Z}) \;\simeq\; \mathbb{Z}

from nnth cohomotopy to nnth integral cohomology is a bijection.

(e.g. Kosinski 93, IX (5.8), Kobin 16, 7.5)


In equivariant homotopy theory

The equivariant Hopf degree theorem – Theorem below – is a generalization of the Hopf degree theorem to equivariant homotopy theory, due to tomDieck 79, 8.4. It implies a fairly explicit characterization of equivariant cohomotopy of representation spheres S VS^V in RO(G)-degree VV (Prop. below).

We need the following list of ingredients and assumptions:

Let GG be a finite group. For HGH \subset G a subgroup, write

(1)W GH(N GH)/H W_G H \coloneqq (N_G H) / H

for its Weyl group.

For XX a G-space, we write

(2)Isotr X(G)Sub(G) Isotr_X(G) \subset Sub(G)

for the subset of the subgroup lattice on the isotropy groups of XX, hence those subgroups which appear as stabilizer subgroups Stab G(x)Stab_G(x) of some point xXx \in X. This means that if H 1,H 2Iso X(G)H_1, H_2 \in Iso_X(G) and H 1<H 2H_1 \lt H_2 is a strict inclusion, then the fixed loci differ X H 1>X H 2X^{H_1} \gt X^{H_2}.

Definition

(matching pair of GG-spaces)

For GG a finite group, we say that a pair X,YGSpacesX,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 HIsotr 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)dim\big( X^H\big) \in \mathbb{N};

  2. H dim(X H)(X H,)H^{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() × 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)\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,)H^{dim(X^H)}\Big( Y^H , \mathbb{Z}\Big) \simeq \mathbb{Z} and hence orientation behaviour (3) e H,Y:W G(H) ×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 =±1H dim(X H)(X H,) or Y,H =±1H dim(X H)(Y H,) \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:XYf \colon X \to Y, we have for each isotropy group HIsotr X(G)H \in Isotr_X(G) that the continuous function f H:X HY Hf^H \;\colon\; X^H \to Y^H has a well-defined integer degree

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

(tom Dieck 79, p. 212 and p. 213)

Example

Let GG be a finite group and VRO(G)V \in RO(G) a finite-dimensional orthogonal linear representation of GG. Then the pair

(XS V,YS V) (X \coloneqq S^V,\; Y \coloneqq S^V)

consisting of two copies of the representation sphere of VV is a matching pair of GG-spaces, according to Def. .

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) HS (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. are evidently satisfied.

Example

Let GG be a finite group which arises as the point group GS/NG \simeq S/N of a crystallographic group SIso(E)S \subset Iso(E) of some Euclidean space EE. Then the pair

(XE/N;YS E) \big( X \coloneqq E/N \,; Y \coloneqq S^E \big)

consisting of

  1. the torus E/NE/N which is the quotient space of EE by the given crystallographic sub-lattice NEN \subset E and equipped with the GG-action descending from that on EE (this Prop.);

  2. the representation sphere of the linear action of the point group GG on EE

is a matching pair of GG-spaces, according to Def. .

Proof

The torus E/NE/N carries the structure of a smooth manifold for which the GG-action is smooth. Since GG is finite, also all its fixed loci (E/N) H(E/N)^H are smooth manifolds (this Prop.). By the equivariant triangulation theorem, all these are WH-CW-complexes.

Moreover, the orientability and connectivity assumptions in Def. are evidently satisfied, using the fact that both E/NE/N as well as S ES^E are modeled on the same linear GG-representation space EE.

Theorem

(equivariant Hopf degree theorem)

Given a matching pair of GG-spaces X,YX, Y (Def. ) 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:XYf \colon X \to Y to its HH-degrees (5)

deg : π 0Maps(X,Y) G AAAA HIsotr X(G) [f] (Hdeg(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 HIsotropy 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)|/|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 ff' 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)deg\big( (f')^H\big) \in \mathbb{Z} allowed by the constraint (6).

(tom Dieck 79, 8.4)

As a special case of the equivariant Hopf degree theorem (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). 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) HGH \underset{\neq}{\subset} G in S VS^V, and a copy of 2\mathbb{Z}_2 or \mathbb{Z} depending on whether V G=0V^G = 0 or not:

π V(S V) {}/ { 2 | V G=0 | otherwise}×HIsotr S V(G)HG|W G(H)| [S VcS V] (Hdeg(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) Hc H(S V) H){ | dim(V H)>0 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)|} \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 ).

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

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 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+VS 1+VS^{1+V} \to S^{1+V} which restrict on S 1S 1S^1 \to S^1 to degree zero or one.

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

(7)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 (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 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 (8) 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 }
Remark

This result

π sgn( sgn) {}/ 2× \pi^{ \mathbb{R}_{sgn} }\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z}_2 \times \mathbb{Z}

from Example

becomes, after stabilization to equivariant stable homotopy theory, the stable homotopy groups of the equivariant sphere spectrum in RO(G)-grading given by

π stab sgn( sgn) {}/× \pi^{ \mathbb{R}_{sgn} }_{stab}\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z} \times \mathbb{Z}

see there.

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.


Applications


References

Due to Heinz Hopf.

Textbook accounts:

Review:

  • Andrew Kobin, Algebraic Topology, 2016 (pdf)

Generalization to equivariant cohomotopy and equivariant cohomology

Last revised on May 30, 2019 at 17:25:08. See the history of this page for a list of all contributions to it.