Bredon cohomology methods in mass partition problems on spheres

BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES SUR OJIT GHOSH Abstract. W e apply R O( G )-graded Bredon cohomology to mass as- signmen t problems, extending classical mass partition metho ds. Within this framework, we reprov e a recen t result of Lessure and Sob er´ on: for n + 1 mass assignments on k -dimensional affine subspaces of R n , there exists a k -subspace containing a sphere that simultaneously bisects all measures. This approac h highligh ts a flexible topological framework with p otential for broader applications. 1. Introduction Mass-partition problems lie at the crossroads of combinatorial geometry , measure theory , and algebraic top ology . The classical ham-sandwich the o- r em [20, 4] asserts that any d finite Borel measures on R d can b e simultane- ously bisected by a single h yperplane, a result pro v ed by Stone and T uk ey [21] via the Borsuk–Ulam theorem. This paradigm, translating top ological symmetry argumen ts in to fair-division results, has inspired a v ast literature. Subsequen t generalizations include the Gr ¨ un baum–Hadwiger–Ramos prob- lem on multiple h yp erplanes [10, 11, 18], the necklace-splitting theorem [1], and v arious extensions inv olving spheres, wedges, or fans [19]. A partic- ularly flexible framework is provided by mass assignments , in tro duced b y Mani-Levitsk a, V re ´ cica, and ˇ Ziv aljevi´ c [17], where measures v ary con tin u- ously with affine subspaces. These problems are t ypically analyzed via the c onfigur ation-sp ac e/test-map scheme , whic h enco des admissible partitions in to a G -space X and searches for G -equiv ariant maps X → S ( V ). Nonex- istence of such maps yields partition results, with obstructions detected via equiv ariant cohomology and index theories [9, 17, 5, 6, 2]. Our approac h applies R O( G )-graded Bredon cohomology for elemen tary ab elian 2-groups G , as dev elop ed b y Hausmann and Sc h w ede [12]. In partic- ular, we analyze the induced map of ˜ H ⋆ G ( S 0 ; F 2 )–mo dules arising from the test map X → S ( V ), where the action of Euler classes pla ys a decisive role in detecting obstructions. Date : F ebruary 26, 2026. 2020 Mathematics Subje ct Classific ation. Primary 55N91, 52A35 secondary 55P91, 55N25. Key wor ds and phr ases. Equiv arian t maps, Measure partitions, RO ( G )-graded Bredon cohomology . 1 2 SUROJIT GHOSH W e show that these to ols can b e brought to b ear on the problem of bi- secting measures b y spheres in affine subspaces. Concretely , we recov er the following recent result of Lessure and Sob er´ on [14]: for n + 1 mass assignments on ℓ -dimensional affine subsp ac es of R n , ther e exists a ℓ -subsp ac e c ontaining a spher e that simultane ously bise cts al l the me asur es. In tuitiv ely , a mass assignment on the ℓ -dimensional affine subspaces of R n is a rule that assigns to each such subspace a measure in a w a y that v aries contin uously with the subspace. A precise form ulation, expressed in terms of sections of an appropriate fibre bundle o v er the space of all ℓ -dimensional affine subspaces, can b e found in [2]. While the theo- rem itself is not new, our approach demonstrates ho w represen tation-graded metho ds naturally enco de the required obstructions and suggest a broader applicabilit y to unresolv ed mass assignment problems. Bey ond recov ering the result of Lessure and Sob er´ on, the main contribu- tion of this pap er is metho dological: w e show that R O( G )-graded Bredon cohomology pro vides a flexible and conceptually unified framework for ana- lyzing mass assignmen t problems. Notation 1.1. (1) F or in tegers n ≥ ℓ ≥ 1, w e denote b y V n,ℓ the real Stiefel manifold consisting of ordered ℓ -tuples of orthonormal v ectors in R n . Explicitly , V n,ℓ = n ( v 1 , . . . , v ℓ ) ∈ ( R n ) ℓ    ⟨ v i , v j ⟩ = δ ij o . It is a smo oth compact manifold of dimension dim V n,ℓ = nℓ − ℓ ( ℓ +1) 2 . (2) W e denote by sgn : C 2 → {± 1 } the sign representation. F or each α = ( α 1 , . . . , α ℓ ) ∈ ( C 2 ) ℓ , define the group homomorphism f α ∈ Hom(( C 2 ) ℓ , C 2 ) , b y f α ( g 1 , . . . , g ℓ ) = α 1 g 1 + · · · + α ℓ g ℓ . The asso ciated one-dimensional real represen tation is giv en b y χ α := sgn ◦ f α , χ α ( g 1 , . . . , g ℓ ) = ( − 1) α 1 g 1 + ··· + α ℓ g ℓ . Ev ery real irreducible represen tation of ( C 2 ) ℓ is of this form. In particular, for the standard basis v ectors e i ∈ ( C 2 ) ℓ w e obtain the c haracters χ i := χ e i , 1 ≤ i ≤ ℓ, corresp onding to the canonical co ordinate pro jections. (3) W e denote b y ⋆ the RO ( G )-grading, and b y ∗ the ordinary in teger grading. Ac kno wledgemen t. The author would lik e to thank the referee for their detailed and p ertinent comments, whic h ha v e helped impro v e the clarit y and presen tation of this manuscript. 2. Preliminaries on Bredon Cohomology Ordinary cohomology theories are defined on ab elian groups and repre- sen ted by sp ectra whose homotopy groups concentrate in degree zero. In the equiv ariant setting, this role is play ed by Mackey functors . W e briefly recall their definition and relation to equiv arian t cohomology (see [15] for details). BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 3 The Burnside c ate gory Burn G is the category whose ob jects are finite G - sets. F or ob jects S, T , the morphism set is the group completion of spans b et w een S and T in the category of finite G -sets. Definition 2.1. A Mackey functor is an additiv e functor M : Burn op G → Ab , from the opp osite Burnside category to abelian groups. In this paper, w e focus on the group ( C 2 ) ℓ , the elemen tary ab elian 2-group of rank ℓ . Example 2.2. F or a G -module M , we define a Mac k ey functor M b y the form ula M ( G/H ) = M H . In particular, we consider the Mack ey functors Z and F 2 , corresp onding to the trivial G -mo dules Z and F 2 , resp ectively . F or a real G -representation V equipp ed with a G -inv ariant inner product, w e define S ( V ) := { v ∈ V | ⟨ v , v ⟩ = 1 } , the unit sphere in V , and S V , the one-p oin t compactification of V . In equiv ariant stable homotop y theory , the V -fold susp ension map X 7→ S V ∧ X is in v ertible. W e refer to [16] for constructing the equiv arian t stable homo- top y category , denoted by Sp G , whic h is the homotopy category of G -sp ectra. Since the V -fold suspension map is inv ertible, one can define S α for α ∈ RO ( G ), the Grothendiec k group completion of the monoid of irre- ducible represen tations of G . This construction induces an RO ( G )-grading on homotopy groups. Sp ecifically , for a based G -space X , the ordinary Bredon cohomology with co efficien ts in the Mack ey functor M at grading α ∈ RO ( G ) is given b y: ˜ H α G ( X ; M ) := Sp G ( X , S α ∧ H M ) , here H M denotes the Eilenberg–Mac Lane spectrum (cf. [13]) for the Mac k ey functor M defined as π n ( H M ) = ( M , n = 0 , 0 , n  = 0 . Let V b e a G -representation with V G = { 0 } . W e denote by a V the inclusion of the fixed p oin ts, a V : S 0 → S V . F or a ring spectrum X with a G -action, w e abuse notation and also denote b y a V its image under the map S 0 → X . If the representation V contains the trivial representation as a summand, then we set a V = 0. Moreo ver, for any tw o G -represen tations V and W , w e ha v e the relation a V a W = a V ⊕ W . 4 SUROJIT GHOSH See [13, Definition 3.11] for further details. F or an orientable G -represen tation V : G → S O ( V ), a choice of orien ta- tion induces an isomorphism ˜ H dim( V ) G ( S V ; Z ) ∼ = Z . The Thom space of the equiv ariant bundle V → G/G is S V . In particular, the restriction map ˜ H dim( V ) G ( S V ; Z ) → ˜ H dim( V ) e ( S dim( V ) ; Z ) is an isomorphism. Utilizing the ab ov e isomorphism, for an orientable G -represen tation V , w e define the orientation class u V as u V ∈ ˜ H V − dim( V ) G ( S 0 ; Z ) , the generator that maps to 1 under the restriction isomorphism. The orien- tation class satisfies the relations: u V ⊕ 1 = u V , u V · u W = u V ⊕ W . See [13, Definition 3.12] for more details. It will b e conv enien t to use the notation G ◦ := Hom( G, C 2 ) \ { 1 } for the set of non trivial characters of G . 3. Equiv ariant formula tion 3.1. Mass partitions as equiv arian t maps. The mass partition problem asks whether measures on R n can b e sim ultaneously bisected b y geometric ob jects such as hyperplanes or spheres. A key insight in [17], is that suc h problems can b e translated into the existence or nonexistence of equiv arian t maps b etw een represen tation spheres and certain spaces. W e no w describ e the equiv ariant framework relev ant for our situation. Consider the action of ( C 2 ) ℓ on the real Stiefel manifold V n,ℓ , given by ( g 1 , . . . , g ℓ ) · ( v 1 , . . . , v ℓ ) = ( g 1 v 1 , . . . , g ℓ v ℓ ) , where each g i ∈ {± 1 } acts on R n b y s calar m ultiplication. Assume, for the sake of con tradiction, that the following statemen t fails: Given n + 1 mass assignments on ℓ -dimensional affine sub- sp ac es of R n , ther e exists a ℓ -dimensional subsp ac e and a spher e c ontaine d in it that simultane ously bise cts al l the me a- sur es. F ollowing [14, page 7], this failure is equiv alen t to the existence of a con tin uous ( C 2 ) ℓ +1 -equiv ariant map S n × V n,ℓ − → S ( V ) , with V = R n ⊕ R n − 1 ⊕ · · · ⊕ R n − ℓ . BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 5 Where the action of ( C 2 ) ℓ +1 on S n × V n,ℓ as the pro duct of the an tip odal action of C 2 on S n and the natural sign-c hange action of ( C 2 ) ℓ on the Stiefel manifold V n,ℓ . F or i = 0 , . . . , ℓ let m i = (1 , . . . , 1 , − 1 , 1 , . . . , 1) ∈ ( C 2 ) ℓ +1 denote the standard generators, where the en try − 1 app ears in the i + 1-st p osition. The action of ( C 2 ) ℓ +1 on V is described as follows. F or ( x 0 , x 1 , . . . , x ℓ ) ∈ V and i = 1 , . . . , ℓ , m i c hanges the sign of every co ordinate of x i and fixes all other comp onen ts. Finally , m 0 c hanges the sign of ev ery co ordinate of x 0 and the first co ordinate of each of x 1 , · · · , x ℓ . Note that, as a ( C 2 ) ℓ +1 -represen tation, V decomp oses as V ∼ = χ ⊕ n 0 ⊕ ℓ M i =1  ( χ 0 ⊗ χ i ) ⊕ χ ⊕ ( n − i − 1) i  , (3.1) where the decomposition is tak en with resp ect to the irreducible characters χ i defined in § 1.1. 3.2. An RO ( G ) -graded spectral sequence. W e no w describ e an R O ( G )- graded Bredon cohomological spectral sequence that helps up to understand the ˜ H ⋆ G ( S 0 ; F 2 )-mo dule structure on the RO ( G )-graded Bredon cohomology of b oth the universal space E G and the pro duct space S n × V n,ℓ . Before we start, note that for any G -space X , the canonical collapsing map X + → S 0 is G -equiv ariant, and therefore induces a ˜ H ⋆ G ( S 0 ; F 2 )-mo dule structure on ˜ H ⋆ G ( X + ; F 2 ). Consider a G -CW complex decomp osition X = S s X ( s ) . If X is free, eac h s -cell is of the form G/e × D s . In this case, the orbit space X/G inherits a CW structure whose s -skeleton is X ( s ) /G , and w e hav e X ( s ) /X ( s − 1) ∼ = _ e ∈ I ( s ) G + ∧ S s . F ollowing [3], one obtains an RO ( G )-graded sp ectral sequence: Prop osition 3.1. F or a fr e e G -CW c omplex X , ther e exists a sp e ctr al se- quenc e E s,t 2 ( α ) = H s  X/G ; π t  S − dim( α ) ∧ H F 2  = ⇒ H s − t − α G ( X ; F 2 ) , with differ entials d r : E s,t r ( α ) − → E s + r,t − r +1 r ( α ) . These sp e ctr al se quenc es assemble (as α varies) into a multiplic ative RO ( G ) - gr ade d sp e ctr al se quenc e E s,α 2 = H s  X/G ; π 0  S − dim( α ) ∧ H F 2  = ⇒ H s − α G ( X ; F 2 ) , wher e s ∈ Z and α ∈ RO ( G ) . 6 SUROJIT GHOSH The E 2 -page is concentrated entirely along the single horizon tal line t = − dim( α ) . F or r ≥ 2, if E s,t r ( α ) is nonzero then necessarily t = − dim( α ), while the target bidegree satisfies t − r + 1  = − dim( α ). Hence the target group is zero, and therefore d r = 0 for all r ≥ 2 . Thus the sp ectral sequence collapses at the E 2 -page. The multiplicativ e structure in Prop osition 3.1 shows that every element can b e written as a pro duct of the form H s ( X/G ) ⊗ π 0  S − dim α ∧ H F 2  . The groups π 0  S − dim( α ) ∧ H F 2  assem ble, as α v aries, into a graded ring of the form O χ ∈ Hom( G,C 2 ) F 2  u ± χ  , hence we hav e: Corollary 3.2. The Br e don c ohomolo gy of E G is ˜ H ⋆ G  E G + ; F 2  ∼ = F 2  x i , u ± χ   χ ∈ G ◦ , 1 ≤ i ≤ ℓ  , with | x i | = 1 . R emark 3.3 . W e iden tify the class a χ i with x i u χ i . F or further details, see [3, page 7]. Since G acts freely on S n × V n,ℓ , the same metho d used in Corollary 3.2 yields: Corollary 3.4. The R O ( G ) -gr ade d Br e don c ohomolo gy of the or der e d c on- figur ation sp ac e S n × V n,ℓ is ˜ H ⋆ G  S n × V n,ℓ ; F 2  ∼ = H ∗ ( R P n × G R (1 , . . . , 1 , n − ℓ ); F 2 ) ⊗ O χ ∈ G ◦ F 2  u ± χ  . 4. Cohomology of cer t ain quotient of O ( n ) In this section, w e in v estigate the cohomology of certain quotient spaces that naturally arise in the study of Stiefel and flag manifolds. W e begin b y recalling classical computations of the cohomology rings of Stiefel manifolds, orthogonal groups, and their classifying spaces. Prop osition 4.1 ([7]) . L et 1 ≤ ℓ ≤ n . Then: (1) H ∗ ( V n,ℓ ; F 2 ) ∼ = F 2 [ z n − ℓ , . . . , z n − 1 ] . (2) H ∗ ( O ( n ); F 2 ) ∼ = F 2 [ y 1 , . . . , y n − 1 ] . (3) H ∗ ( B O ( n ); F 2 ) ∼ = F 2 [ ω 1 , . . . , ω n ] . BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 7 4.1. Flag Manifolds ov er R . Let n 1 , . . . , n s b e p ositiv e in tegers with n 1 + · · · + n s = n . An ( n 1 , . . . , n s ) -flag over R is an ordered collection σ = ( σ 1 , . . . , σ s ) of mutually orthogonal subspaces of R n , where dim σ i = n i . The space of all suc h flags is a compact smo oth manifold called the r e al flag manifold , denoted G R ( n 1 , . . . , n s ). R emark 4.2 . The case G R ( n 1 , n 2 ) is the Grassmannian of n 1 -planes in R n 1 + n 2 , i.e. G R ( n 1 , n 2 ) ∼ = Gr n 1 ( R n 1 + n 2 ). The real flag manifold can b e described as the homogeneous space G R ( n 1 , n 2 , . . . , n s ) ∼ = O ( n ) O ( n 1 ) × O ( n 2 ) × · · · × O ( n s ) . The group O ( n ) acts transitiv ely on flags, with stabilizer the blo ck-diagonal subgroup O ( n 1 ) × · · · × O ( n s ), giving G R ( n 1 , . . . , n s ) its natural smo oth structure and an O ( n )-inv ariant Riemannian metric. Note that V n,ℓ ∼ = O ( n ) O ( n − ℓ ) and the group ( C 2 ) ℓ = O (1) ℓ acts freely on it as describ ed in § 3.1. The quotien t map V n,ℓ − → G R (1 , . . . , 1 , n − ℓ ) is therefore a principal ( C 2 ) ℓ –bundle. Consequen tly , it is classified by a map to the classifying space B ( C 2 ) ℓ ≃ ( RP ∞ ) ℓ , and w e obtain a homotop y fibration V n,ℓ − → G R (1 , . . . , 1 , n − ℓ ) − → ( RP ∞ ) ℓ , (4.1) 4.2. The sp ectral sequence computation. W e apply the Serre sp ectral sequence in cohomology (with coefficients in a field F 2 ) to the fibration (4.1). The E 2 -page has the form E p,q 2 = H p  ( RP ∞ ) ℓ ; H q ( V n,ℓ ; F 2 )  = ⇒ H p + q ( G R (1 , . . . , 1 , n − ℓ ); F 2 ) , where H q ( V n,ℓ ; F 2 ) denotes the lo cal co efficient system on the base induced b y the mono drom y action of π 1  ( RP ∞ ) ℓ  on H q ( V n,ℓ ; F 2 ). The fundamen tal group π 1  ( RP ∞ ) ℓ  ∼ = ( Z / 2) ℓ acts on V n,ℓ b y sign c hanges in eac h coordinate: ( λ 1 , . . . , λ ℓ ) · ( v 1 , . . . , v ℓ ) = ( v 1 , . . . , v ℓ ) · diag ( λ 1 , . . . , λ ℓ ) , where each λ i ∈ {± 1 } . Over F 2 -co efficien ts, this action is trivial on cohomol- ogy . Therefore the lo cal system H q ( V n,ℓ ; F 2 ) is constant, and the E 2 -page splits as a tensor pro duct: E p,q 2 ∼ = H p  ( RP ∞ ) ℓ ; F 2  ⊗ F 2 H q ( V n,ℓ ; F 2 ) . Note that the cohomology of ( RP ∞ ) ℓ is a p olynomial algebra H ∗  ( RP ∞ ) ℓ ; F 2  ∼ = F 2 [ x 1 , . . . , x ℓ ] , 8 SUROJIT GHOSH where x i = w 1 ( L i ) ∈ H 1 ( RP ∞ ; F 2 ) is the first Stiefel–Whitney class of the tautological line bundle L i o v er the i -th factor. The cohomology of V n,ℓ is generated by classes z n − ℓ , z n − ℓ +1 , . . . , z n − 1 in degrees n − ℓ, n − ℓ + 1 , . . . , n − 1, resp ectiv ely . Thus E p,q 2 ∼ = F 2 [ x 1 , . . . , x ℓ ] ⊗ F 2 F 2 [ z n − ℓ , . . . , z n − 1 ] . Consider the following homotopy commutativ e diagram of fibrations ( A ) , ( B ) , ( C ), and ( D ): ( A ) ( B ) ( C ) ( D ) V n,ℓ   V n,ℓ   O ( n )   ϕ o o O ( n )   G R (1 , . . . , 1 , n − ℓ )   / / O ( n ) O ( ℓ ) × O ( n − ℓ )   O ( n ) O ( ℓ ) × O ( n − ℓ )   / / E O ( n )   B O (1) ℓ g / / B O ( ℓ ) B O ( ℓ ) × B O ( n − ℓ ) π 1 o o f / / B O ( n ) . W e analyze the induced Serre sp ectral sequences and their differentials using naturality . Lets b egin with the univ ersal fibration ( D ), O ( n ) − → E O ( n ) − → B O ( n ) . It is classical that the transgression in the Serre sp ectral sequence satisfies τ ( y i ) = w i +1 , where w i +1 is the ( i + 1)-st Stiefel–Whitney class of the univ ersal n -plane bundle ov er B O ( n ). Next, consider the fibration ( C ). The map f : B O ( ℓ ) × B O ( n − ℓ ) − → B O ( n ) classifies the v ector bundles ξ ℓ × ξ n − ℓ , where ξ k denotes the canonical k - plane bundle o ver B O ( k ). By naturalit y of the Serre sp ectral sequence with resp ect to the map f , the differential in ( C ) is giv en by d i +1 ( y i ) = w i +1 ( ξ ℓ × ξ n − ℓ ) ∈ H i +1 ( B O ( ℓ ) × B O ( n − ℓ ); F 2 ) . Let w i = w i ( ξ ℓ ), ˜ w i = w i ( ξ n − ℓ ), and write w = 1 + w 1 + · · · + w ℓ , w ′ = 1 + w ′ 1 + · · · for the in v erse of w . By Prop osition 11.1 of [8], we ha v e X i + j = r ˜ w i w j = 0 for all r ≤ n − ℓ. BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 9 Comparing this with (1 + w 1 + · · · + w ℓ )(1 + w ′ 1 + · · · ) = 1 sho ws that ˜ w i = w ′ i for all i ≤ n − ℓ . Applying this on the E n − ℓ +1 -page giv es d n − ℓ +1 ( y n − ℓ ) = w n − ℓ +1 ( ξ ℓ × ξ n − ℓ ) = X i + j = n − ℓ +1 i ≤ ℓ, j ≤ n − ℓ ˜ w i w j = X i + j = n − ℓ +1 i ≤ ℓ, j ≤ n − ℓ w i w ′ j = − w ′ n − ℓ +1 . The same argumen t yields d j +1 ( y j ) = − w ′ j +1 for all j ≥ n − ℓ . No w consider the fibration ( B ). The map ϕ : O ( n ) − → V n,ℓ induces a pullbac k on cohomology . A direct insp ection of the generators sho ws that y 1 , . . . , y n − ℓ − 1 / ∈ Im( ϕ ∗ ) , while for j ≥ n − ℓ we hav e z j = ϕ ∗ ( y j ) . Comparing the Serre sp ectral sequences for ( B ) and ( C ) yields d j +1 ( z j ) = − w ′ j +1 j ≥ n − ℓ. This completely determines the non trivial differen tials in the Serre sp ec- tral sequence asso ciated to the fibration ( B ). W e no w determine the differen tials in the Serre spectral sequence asso ci- ated to the fibration ( A ), V n,ℓ − → G R (1 , . . . , 1 , n − ℓ ) − → B O (1) ℓ . Note that g : B O (1) ℓ → B O ( ℓ ) is the classifying map for the Whitney sum γ 1 ⊕ · · · ⊕ γ ℓ . Hence g ∗ ( w ′ ) = Q ℓ i =1 (1 + x i ) − 1 . Consequen tly , for j = n − ℓ, · · · , n − 1, the differen tial (for the sp ectral sequence ( A )) is given by d j +1 ( z j ) = − g ∗ ( w ′ j +1 ) = f j +1 ( x 1 , . . . , x ℓ ) (mo d 2) , where f j +1 denotes the complete symmetric p olynomial of degree j + 1. All other differentials v anish for degree reasons. These relations com- pletely determine the E ∞ -page and hence we hav e Prop osition 4.3. The c ohomolo gy ring of the r e al flag manifold satisfies H ∗ ( G R (1 , . . . , 1 , n − ℓ ); F 2 ) ∼ = F 2 [ x 1 , . . . , x ℓ ] / ( f n − ℓ +1 , . . . , f n ) , wher e f j denotes the c omplete symmetric p olynomial of de gr e e j in x 1 , . . . , x ℓ . R emark 4.4 . The top non trivial cohomology class of G R (1 , · · · , 1 , n − ℓ ) is ℓ Y i =1 x n − i i , whic h generates the one-dimensional top cohomology group. 10 SUROJIT GHOSH 5. The Pr oof W e b egin by recalling some k ey structural facts ab out represen tation- graded Bredon cohomology rings due to Hausmann-Sch w ede [12]. These results will pro vide the algebraic foundation for all subsequen t arguments. Let e H Rep G ( S 0 ) denote the R O ( G )-graded reduced Bredon cohomology of S 0 consisting of those degrees of the form V − k , where V is a finite- dimensional real G -representation and k ∈ Z . Note that in [12] it is denoted b y H ( G, ⋆ ) . In our notation H m ( G, V ) corresp onds to e H V − m G ( S 0 ; F 2 ). Prop osition 5.1 ([12, Prop osition 2.1]) . L et G b e an elementary ab elian 2 -gr oup and let V b e a G -r epr esentation with trivial fixe d p oints. (1) F or every sub gr oup H ≤ G , the r estriction homomorphism res G H : e H V − m G ( S 0 ; F 2 ) − → e H V − m G ( G/H + ; F 2 ) is surje ct ive. (2) Supp ose χ is a nontrivial G -char acter with kernel K . Then the fol- lowing se quenc e is exact: 0 → e H W − m G ( S 0 ; F 2 ) · a χ → e H W ⊕ χ − m G ( S 0 ; F 2 ) res G K → e H W ⊕ χ − m G ( G/K + ; F 2 ) → 0 . (3) The F 2 -ve ctor sp ac e e H V − m G ( S 0 ; F 2 ) is sp anne d by the classes a U · u W for al l G -r epr esentations U and W such that U ⊕ W = V and m = dim( W ) . Theorem 5.2 ([12, Theorem 2.2]) . F or every elementary ab elian 2 -gr oup G , the r epr esentation-gr ade d Br e don homolo gy ring ˜ H Rep G ( S 0 ; F 2 ) is an inte gr al domain. Prop osition 5.3. L et G b e an elementary ab elian 2-gr oup and V a G - r epr esentation with trivial fixe d p oints. Then ˜ H V G ( S ( V ) + ; F 2 ) ∼ = 0 . Pr o of. Consider the cofibration sequence of based G -spaces S ( V ) + i − → S 0 j − → S V , where S ( V ) + denotes the unit sphere S ( V ) with a disjoin t basep oint. Applying the reduced representation-graded Bredon cohomology functor ˜ H V G ( − ; F 2 ), we get the long exact sequence · · · → ˜ H V G ( S V ; F 2 ) j ⋆ − → ˜ H V G ( S 0 ; F 2 ) i ⋆ − → ˜ H V G ( S ( V ) + ; F 2 ) → ˜ H V +1 G ( S V ; F 2 ) → · · · . Since a V is not a zero divisor (as ˜ H Rep G ( S 0 ; F 2 ) is a domain by Theorem 5.2) and the map j ⋆ corresp onds to multiplication b y a V , th us, j ⋆ is injectiv e, BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 11 and th us i ⋆ is the zero since ˜ H V G ( S 0 ; F 2 ) ∼ = F 2 b y Proposition 5.1 (see part (3) for m = 0). Consequently , ˜ H V G ( S ( V ) + ; F 2 ) ∼ = 0 , as claimed. □ W e no w turn to the description of the ˜ H ⋆ G ( S 0 ; F 2 )–mo dule structure in the case S n × V n,ℓ , with G = ( C 2 ) ℓ +1 . Prop osition 5.4. L et G = ( C 2 ) ℓ +1 and X = S n × V n,ℓ . Then ˜ H ⋆ G ( X + ; F 2 ) is an ˜ H ⋆ G ( S 0 ; F 2 ) -mo dule with action: (1) F or e ach nontrivial χ ∈ G ◦ , the class u χ ∈ ˜ H ⋆ G ( S 0 ; F 2 ) acts by multiplic ation w ith the c orr esp onding unit u χ . (2) The Euler class a χ acts by multiplic ation with x χ u χ , wher e x χ ∈ H 1 ( B G ; F 2 ) is the de gr e e-one class asso ciate d to χ . Pr o of. Since G acts freely on X , there exists (up to G -homotopy) a classi- fying map ϕ : X → E G . The collapse map factors as X + ϕ + − − → E G + q − → S 0 , so the ˜ H ⋆ G ( S 0 ; F 2 )-action on ˜ H ⋆ G ( X + ; F 2 ) is determined by the pullbac k ϕ ⋆ + . By Corollary 3.2, ˜ H ⋆ G ( E G + ; F 2 ) is generated by in verti ble Thom classes u χ and Euler classes a χ = x χ u χ . Pulling back giv es ϕ ⋆ + ( u χ ) = u χ . F or Euler classes we obtain ϕ ⋆ + ( a χ ) = ϕ ⋆ + ( x χ ) u χ , where ϕ ⋆ + ( x χ ) is the natural degree-one class on X/G : for χ = χ 0 it is the generator x 0 ∈ H 1 ( R P n ; F 2 ), and for χ = χ i with 1 ≤ i ≤ ℓ it is x i from the partial flag factor as in Prop osition 4.3. This yields the stated formulas. □ Corollary 5.5. L et V b e as in (3.1) . Then the action of a V on e H ⋆ G ( S n × V n,ℓ ; F 2 ) is given by multiplic ation by the element x n 0 u n 0 · ℓ Y i =1 x n − i i · u χ 0 ⊗ χ i u n − i − 1 χ i . Pr o of. The class a V is computed m ultiplicativ ely from the Euler classes of the summands in the decomp osition of V . More precisely , a V = a n χ 0 ℓ Y i =1 a χ 0 ⊗ χ i a n − i − 1 χ i , where a χ i denotes the Euler class of the real line bundle asso ciated with the c haracter χ i . Since the first Stiefel–Whitney class is additive under tensor pro ducts of line bundles, w e hav e a χ 0 ⊗ χ i = a χ 0 + a χ i . The claim no w follo ws directly from Prop osition 5.4 and the fact that x n +1 0 = 0. □ 12 SUROJIT GHOSH Theorem 5.6. Ther e do es not exist any ( C 2 ) ℓ +1 -e quivariant map S n × V n,ℓ − → S ( V ) , with V as describ e d in (3.1) . Pr o of. Consider G = ( C 2 ) ℓ +1 . Ass ume for contradiction that suc h a map f exists. Using the ˜ H ⋆ G ( S 0 ; F 2 )-mo dule structure, we obtain the following comm utativ e diagram: ˜ H 0 G  S ( V ); F 2  f ∗ / / a V .   ˜ H 0 G  S n × V ℓ ( R d ); F 2  a V .   ˜ H V G  S ( V ) + ; F 2  f ⋆ / / ˜ H V G  S n × V ℓ ( R d ); F 2  By Prop osition 5.3, we yield: 0 = f ⋆ ( a V . 1) = a V .f ∗ (1) = x n 0 u n 0 · ℓ Y i =1 x n − i i · u χ 0 ⊗ χ i u n − i − 1 χ i . whic h is non-trivial b y Corollary 3.4 and Remark 4.4. This con tradicts our assumption, hence the result follo ws. □ References [1] N. Alon, Splitting ne cklac es, Adv. Math. 63 (1987), 247–253. [2] I. Axelrod-Freed, P. Sober ´ on, Bise ctions of mass assignments using flags of affine spac es, Discrete Comput. Geom. (2022), 19 pp. (online first: 30 December 2022). [3] S. Basu, S. Ghosh, Br e don c ohomolo gy of finite dim ensional C p -sp ac es, Homology Homotop y Appl. 23 (2021), no. 2, 33–57. [4] W. A. Beyer, A. Zardecki, The e arly history of the ham sandwich the or em, Amer. Math. Monthly 111 (2004), no. 1, 58–61. [5] P. V. M. Blagojevi ´ c, F. Frick, A. Haase, G. M. Ziegler, Hyp erplane mass p artitions via r elative e quivariant obstruction the ory, Do c. Math. 21 (2016), 735–771. [6] P. V. M. Blagojevi ´ c, F. Frick, A. Haase, G. M. Ziegler, T opolo gy of the Gr¨ unb aum–Hadwiger–R amos hyp erplane mass p artition pr oblem, T rans. Amer. Math. So c. 370 (2018), no. 10, 6795–6824. [7] A. Borel, Sur la c ohomolo gie des esp ac es fibr ´ es princip aux et des esp ac es homo g` enes de gr oup es de Lie c omp acts, Ann. of Math. (2) 57 (1953), 115–207. [8] A. Borel, L a c ohomolo gie mo d 2 de c ertains esp ac es homo g` enes, Comment. Math. Helv. 27 (1953), 165–197. [9] E. F adell, S. Husseini, An ide al-value d c ohomolo gic al index the ory, Michigan Math. J. 35 (1988), no. 3, 319–328. [10] B. Gr ¨ unbaum, Partitions of mass-distributions and of c onvex b o dies by hyp erplanes, P acific J. Math. 10 (1960), 1257–1261. [11] H. Hadwiger, Simultane Vierteilung zweier K¨ orp er, Arch. Math. (Basel) 17 (1966), 274–278. [12] M. Hausmann, S. Schwede, R epr esentation-gr aded Br e don homolo gy of elemen- tary ab elian 2-gr oups, arXiv preprint arXiv:2403.05355 [math.A T], 2024. Av ailable at: BREDON COHOMOLOGY METHODS IN MASS P AR TITION PR OBLEMS ON SPHERES 13 [13] M. A. Hill, M. J. Hopkins, D. C. Ra venel, On the nonexistenc e of elements of Kervair e invariant one, Ann. of Math. (2) 184 (2016), no. 1, 1–262. [14] D. Lessure, P. Sober ´ on, Partitions of mass assignments with spher es and we dges, arXiv preprint arXiv:2507.06919, 2025. [15] Ma y, Peter J. “Equiv arian t homotopy and cohomology theory .” V ol. 91 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, W ashington, DC; b y the American Mathematical Society , Pro vidence, RI, 1996. With contributions by M. Cole, G. Comezan˜ a, S. Costenoble, A. D. Elmendorf, J. P . C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. T riantafillou, and S. W aner. [16] Mandell, M. A.; Ma y, J. P. “Equiv arian t orthogonal spectra and S -mo dules.” Mem. Amer. Math. So c. 159 (2002), no. 755, x+108 pp. [17] P. Mani-Levitska, S. Vre ´ cica, R. ˇ Ziv aljevi ´ c, T op olo gy and c ombinatorics of p artitions of masses by hyp erplanes, Adv. Math. 207 (2006), no. 1, 266–296. [18] E. A. Ramos, Equip artition of mass distributions by hyp erplanes, Discrete Comput. Geom. 15 (1996), no. 2, 147–167. [19] E. R old ´ an-Pensado, P. Sober ´ on, A survey of mass p artitions, (2020). [20] H. Steinhaus, A note on the ham-sandwich theor em, Mathesis Polsk a (1938). [21] A. H. Stone, J. Tukey, Gener alize d “sandwich” the or ems, Duke Math. J. 9 (1942), 356–359. Dep ar tment of Ma thema tics, Indian Institute of Technology, Roorkee, Utt arakhand-247667, India Email addr ess : surojit.ghosh@ma.iitr.ac.in; surojitghosh89@gmail.com