Basic equivariant Thom isomorphism

since ±id is an isomorphism, the 5-lemma (cf. [ES15, Lemma 4.3]) then yields that ϕ is an isomorphism, as well. First, we determine the boundary operator∂. Letθ represent a class inH_G^k−1(A,E). By definition of the long exact cohomology sequence, ∂θ is determined as follows: ι^∗_A : C_G^k(N,E) → C_G^k(A,E) is surjective, so there is a form ω ∈ C_G^k(N,E) : ι^∗_Aω = θ. But ι^∗_Ad_Gω = d_Gθ = 0, hence, dω ∈ker(ι^∗_A) = im(ι^∗) and there existsγ ∈He_G^k(N, A,E) with ι^∗γ =d_Gω. ι^∗d_Gγ = d_Gd_Gω = 0, so we have d_Gγ = 0 by injectivity of ι^∗. Then ∂θ := γ. Now, let π : U → A denote an invariant tubular neighborhood and f : N → R an invariant function with suppf ⊂ U and f|_Ue ≡ 1 on a smaller neighborhood of A. Then ω := f π^∗θ extends θ to N. Since d_Gθ = 0, we have d_Gω = df ∧π^∗θ.

(df ∧π^∗θ)|_A = 0 since df|_Ue = 0, hence, df ∧π^∗θ ∈ He_G^k(N, A,E). It follows that

∂θ =df∧π^∗θ. Further, we haveD(f π^∗θ,0) = (df∧π^∗θ,(f π^∗θ)|_A) = (df∧π^∗θ, θ), so(df∧π^∗θ,0)and(0,−θ)represent the same relative cohomology class. It follows that α^∗(−id(θ)) = (0,−θ) = (df ∧π^∗θ,0) =ϕ◦∂(θ), the diagram commutes.

The basic equivariant pushforwardi∗will be constructed as follows. Anequivariant basic Thom form is a closed formτ ∈C_G,c^d (U,F)satisfyingp∗τ = 1. We will give a construction of equivariant basic Thom forms at the end of this section. Suppose for now that an equivariant basic Thom form has been constructed. Then we define the basic equivariant pushforward as the composition

i∗ :C_G^k(A,F) ^p

∗

→C_G^k(U,F)→^∧τ C_G,c^k+d(U,F)→C_G^k+d(M,F), (3.5) where the last arrow denotes extension by zero.

Proposition 3.4.2. The basic equivariant pushforward satisfies, for all closed forms β ∈C_G(A,F) and η∈C_G(U,F)

Z

M

η∧i∗β = Z

A

i^∗η∧β.

Proof. i∗β =p^∗β∧τ is a form compactly supported in an invariant neighborhood U of A. Therefore we have

Z

M

η∧i∗β = Z

U

η∧p^∗β∧τ (by definition of i∗)

= Z

U

p^∗i^∗η∧p^∗β∧τ (by Proposition 3.3.10)

= Z

A

i^∗η∧β∧p∗τ (by Lemma 3.4.1)

= Z

A

i^∗η∧β (by p∗τ = 1).

We obviously havep_∗◦i_∗ = id. Analogously to [GS99, Theorem 10.6.1], we obtain for the induced maps on cohomology:

Theorem 3.4.3(Basic equivariant Thom isomorphism).Integration over the fiber defines an isomorphism

p∗ :H_G,c^k (U,F)→H_G^k−d(A,F) whose inverse is given by i_∗.

It remains to construct the equivariant basic Thom form. We use a variant of the Mathai-Quillen construction based on the presentation in [GS99, Chapter 10] (see

also [Töb14, GNT17] for closely related constructions). First we identify U with the normal bundle νA→A, equipped with a G×T-invariant metric. Let P →A denote the bundle of oriented orthonormal frames ofνA: it is a G×T-equivariant principal SO(d)-bundle over A. Consider the map P ×R^d→νA,

(x,(e₁, . . . , e_d), v)→(x, v₁e₁+· · ·+v_de_d).

It gives aG×T-equivariant diffeomorphism(P ×R^d)/SO(d)∼=νA. EquipP with a G×T-invariant basic connection form. Recall that such a form has to exist, see Remark 2.5.3. Using the Cartan model of equivariant basic cohomology, the Cartan map yields isomorphisms

φ_νA:C_SO(d)×G,c(P ×R^d,E)→^∼= C_G,c(νA,F) φ_A:CSO(d)×G(P,E × {∗})→^∼= C_G(A,F),

where E denotes the foliation induced by R on P. Let p₂ : P ×R^d → R^d be the projection. We define τ by

τ :=φ_νA(p^∗₂(ν⊗1)) ∈C_G,c(νA,F)

where ν ∈ C_SO(d),c(R^d) is the (modified) universal Thom-Matthai-Quillen form as constructed in [GS99, Section 10.3], ν ⊗1 ∈ C_SO(d)×G,c(R^d). By analogous arguments to [GS99, Section 10.4], we hence have the following.

Theorem 3.4.4. The form τ ∈ C_G,c^d (U,F) as constructed above is a Thom form for the projection p : U → A. Consequently, the basic equivariant pushforward i_∗ :H_G^k(A,F)→H_G^k+d(M,F) is well-defined.

In Section 2.3, we scrutinized the functionsΨ^Xs+1|_Ys and the connected components N of their critical sets. Recall that everyN is aG×T-invariant closed submanifold of even codimension (cf. Lemma 2.4.8) and non-degenerate (cf. Proposition 2.4.9).

We will now consider the special case that A = N. Denote the Morse index of Ψ^Xs+1|_Ys on N by λ, the inclusion as the zero section N → ν^±N by ι^± and the projection by p^± : ν^±N → N. For the following definition, compare [GNT17, Section A.1].

Definition 3.4.5. Letk denote the rank of the (positive/negative) normal bundle ν^(±)N. Then the bundle P of oriented orthonormal frames of (ν^(±)N,F) is a

foliated SO(k)-bundle over N. The equivariant basic Euler form e_G(ν^(±)N,F) ∈ C_G^∗(N,F) of (ν^(±)N,F)→(N,F)is defined by

e_G(ν^(±)N,F)(X) = Pf(F_G^θ(X)) = Pf(F^θ−ι_Xθ),

where θ ∈ Ω¹(P,F)^G ⊗ so(k) denotes a G-invariant basic connection form on the bundle of oriented orthonormal frames of (ν^(±)N,F), F_G^θ = d_Gθ + ¹₂[θ, θ] its equivariant curvature form and Pf the Pfaffian.

For any G× {ψ_t}-invariant connection form, we can analogously define the equi-variant Euler formeg×RR(ν^(±)N)∈Cg×RR(N)or, for aG×T-invariant connection form, the equivariant Euler form eG×T(ν^(±)N)∈CG×T(N).

Note that, while the Eulerform depends of the choice of connection form, its class (for which we use the same notation) does not. We can think of eg×RR(ν^(±)N) as the restriction of the polynomial map eG×T(ν^(±)N)to g×RR.

Proposition 3.4.6. Under the S(g^∗)-algebra isomorphism H_G(N,F) = Hg×RR(N) of Proposition 3.3.3, e_G(ν^(±)N,F) = eg×RR(ν^(±)N).

Proof. e_G(ν^(±)N,F) is, by definition, computed with respect to a G-invariant R-basic connection form θ. Then θ is, in particular, G× {ψ_t}-invariant and satisfies θ(R) = 0 so that we can compute eg×RR(ν^(±)N)with respect to the same connec-tion form and obtaineg×RR(ν^(±)N)(R) = e_G(ν^(±)N,F)(0)andeg×RR(ν^(±)N)(X) = e_G(ν^(±)N,F)(X)for every X ∈g. Since the isomorphism H_G(N,F) = Hg×RR(N) is induced by the natural inclusion of complexes

(C_G^∗(N,F), d_G),→

(S(g^∗)⊗S((RR)^∗)⊗Ω^∗(N))^g×RR, dg×RR

, we obtain the claim.

Analogously to Theorem 3.4.3, we obtain

Theorem 3.4.7(Basic equivariant Thom isomorphism).Integration over the fiber defines an isomorphism

p⁻_∗ :H_G,c^∗+λ(ν⁻N,F)→H_G^∗(N,F)

whose inverse is given by the composition ι⁻_∗ :H_G^∗(N,F) ^(p

−)^∗

−−−→H_G^∗(ν⁻N,F)−→^∧τ H_G,c^∗+λ(ν⁻N,F).

As in [GS99, Section 10.5], it can be shown that (ι⁻)^∗τ =e_G(ν⁻N,F) and, hence, that (ι⁻)^∗ι⁻_∗ =∧e_G(ν⁻N,F) is the multiplication with the basic equivariant Euler class of ν⁻N.

The analogous statements hold for the positive and the whole normal bundle, with λ replaced by rank(νN)−λ and rank(νN), respectively.

Basic Kirwan Surjectivity for K -Contact Manifolds

In this Chapter, we state and prove our basic Kirwan surjectivity result. After-wards, we present examples and establish that in the case where the Reeb vector field induces a freeS¹-action, our result reproduces the known Kirwan surjectivity for the S¹-quotient. A Tolman-Weitsman type description of the kernel of the basic Kirwan map for S¹-actions is derived in Section 4.3, for which we then also present an example. In that section, we also obtain an injectivity statement that corresponds to the well-known Kirwan injectivity.

4.1 Basic Kirwan Surjectivity

We will now proceed to state and prove our surjectivity result.

Theorem 4.1.1. Let (M, α) be a compact K-contact manifold, R its Reeb vector field and F the foliation that is induced by R. Let G be a torus that acts on M, preserving α. Denote by Ψ : M →g^∗ the contact moment map and suppose that 0 is a regular value of Ψ. Then the inclusion Ψ⁻¹(0) ⊂M induces an epimorphism in equivariant basic cohomology

H_G^∗(M,F)−→H_G^∗(Ψ⁻¹(0),F).

51

Proof. Choose a metricg adapted toαaccording to Lemma 2.4.7. Let(X₁, ..., X_r) be a basis of g according to Proposition 2.4.1. Let again f_s := (Ψ^X1, ...,Ψ^Xs) : M → R^s, Y₀ := M and Y_s := f_s⁻¹(0) for s = 1, ..., r. By Proposition 2.4.9, the functionsΨ^Xs+1|_Ys are Morse-Bott functions. We will show that the restrictions to the subsets Y_s+1 ⊂Y_s induce the following sequence of epimorphisms:

H_G^∗(M,F) =H_G^∗(Y₀,F)→H_G^∗(Y₁,F)→...→H_G^∗(Y_r,F) = H_G^∗(Ψ⁻¹(0),F).

Set Y_s^c := Ψ^Xs+1|_Ys−1

((−∞, c]). Let κ be a critical value of Ψ^Xs+1|_Ys. We denote the connected components of the critical set at level κ by B₁^κ, ..., B_j^κ_κ and by λ^κ_i the indices of the non-degenerate critical submanifolds B_i^κ with respect to Hess(Ψ^Xs+1|_Ys)and withν^±B_i^κ their positive (respective negative) normal bundles.

Letbe small enough such that the interval[κ−, κ+]contains no critical values of Ψ^Xs+1|_Ys besides κ. Since Ψ^Xs+1|_Ys is a G×T-invariant Morse-Bott function, Y_s^κ+ is (G×T)-equivariantly diffeomorphic to Y_s^κ− with j_κ handle bundles of type(ν⁺B_i^κ, ν⁻B_i^κ) disjointly attached by Theorem A.4.

Y_s^κ+ 'Y_s^κ−∪S

D^ν+Bκi⊕S^ν−Bκi

[D^ν+Biκ⊕D^ν−Bκi. (4.1) Here, the G×T-action onν^±B_i^κ is the natural lift of theG×T-action onM. We denote the foliation induced byR on the normal bundle also byF. LetU_i^κ denote an invariant tubular neighborhood of D^ν+Bκi ⊕D^ν−Biκ. By Diffeomorphism (4.1) and Proposition 3.3.15, we have

H_G^∗(Y^κ+_s , Y_s^κ−,F) = H_G^∗(Y_s^κ−∪_∪Dν+Bκ

i⊕S^ν−Bκi D^ν+Biκ⊕D^ν−Bκi, Y_s^κ−,F)

=H_G^∗(∪ U_i^κ,∪ U_i^κ\D^ν+Biκ⊕D˚^ν−Bκi,F) (by Proposition 3.3.14)

=H_G^∗(∪ D^ν+Bκi ⊕D^ν−Biκ,∪ D^ν+Biκ⊕S^ν−Bκi,F) (by Proposition 3.3.15)

=H_G^∗(∪ D^ν−Biκ,∪S^ν−Biκ,F) (by Proposition 3.3.15)

=M

H_G^∗(D^ν−Biκ, S^ν−Bκi,F)

=M

He_G^∗(D^ν−Biκ, S^ν−Bκi,F) (by Proposition 3.3.21)

=M

H_G,c^∗ ( ˚D^ν−Biκ,F) (by Proposition 3.3.20) Consider theG×T-equivariant diffeomorphismρ: ˚D^ν−Biκ →ν⁻B_i^κ, v 7→ _1−||v||¹ 2v.

Since ρis proper, Proposition 3.3.10 yields H_G^∗(Y_s^κ+, Y_s^κ−,F) = M

H_G,c^∗ (ν⁻B_i^κ,F). (4.2)

By the Thom isomorphism (Theorem 3.4.7), we have further

H_G^∗−λκi(B_i^κ,F)−→^∼ H_G,c^∗ (ν⁻B_i^κ,F). (4.3) With (G× {ψ_t})_B^κ

i we denote the isotropy group of G× {ψ_t} on B_i^κ. Since Ψ and g are G×T-invariant,(G× {ψ_t})_B^κ

i acts fiberwise on ν⁻B^κ_i (and D^λκiB_i^κ) by restriction of the isotropy representation. We need the following lemmata.

Lemma 4.1.2. ν^(±)B_i^κ has no non-zero (G× {ψ_t})_B^κ

i-fixed vectors.

Proof. For x ∈ B_i^κ, let γv be the unique geodesic with initial values γv(0) = x,γ˙v(0) = v, v ∈ νx^(±)B_i^κ. Since G× {ψt} acts by isometries, g ·γv is again a geodesic and, by uniqueness, g·γv =γ_dg(v) for all g ∈(G× {ψt})B_i^κ. Assume v to be a (G× {ψt})B_i^κ-fixed vector. Then g·γv =γ_dg(v) =γv for allg ∈(G× {ψt})B^κ_i, hence the isotropy group of all points alongγv contains (G× {ψt})B_i^κ. By Lemma 2.4.6, however, the critical set is the union of all minimal G× {ψt}-orbits, henceγv

lies completely in the connected component B_i^κ. Thus v = ˙γ_v(0)∈T_xB_i^κ ⊥ν_xB_i^κ, therefore v = 0.

It follows that ν^±B_i^κ has no non-zero (G×T)_B^κ

i-fixed vectors, therefore, the mul-tiplication with the Euler classes of the negative, positive or whole normal bundle in H_G×T^∗ (B_i^κ) is injective (see [Duf83, Proposition 5] or [AB83, Section 13]). We now show that this also holds for their restriction to g×RR.

Lemma 4.1.3. Multiplication in H_g⊕^∗

RR(B_i^κ) with the equivariant Euler class of the negative, positive or whole normal bundle of B_i^κ is injective.

Proof. We present the proof for the case of the negative normal bundle, the other cases work analogously. Denote the Euler class of ν⁻B_i^κ by E_i^κ. Let θ denote a G×T-invariant connection 1-form in the bundleP of oriented orthonormal frames of the negative normal bundle overB_i^κ. Then, by definition, forX ∈g⊕RR,E_i^κ(X) is given by Pf(F^θ−ι_Xθ), where we again denote the Pfaffian ∈ S(so(λ^κ_i)^∗)^SO(λκi) by Pf. The classification of irreducible torus representations yields that ν⁻B_i^κ splits into 2-dimensional subbundles s.t., when written in a basis adapted to the

splitting, the(g⊕t)_B^κ

i-action is given by the standard action of the matrix









0 −α1(X) α1(X) 0

...

0 −α_λκ

i/2(X) α_λκ

i/2(X) 0









, X ∈(g⊕t)B^κ_i (4.4)

with the weights α₁, ..., α_λ^κ

i/2 of the(g⊕t)_B^κ

i-representation. For every X ∈(g⊕ t)_B^κ

i, Matrix (4.4) is an element of so(λ^κ_i). Thus, X_P and the SO-fundamental vectorfield generated by Matrix (4.4) coincide. By the definition of a connection form, θ(Y_P) = Y for every Y ∈ so. Therefore, it holds for every X ∈ (g⊕t)_B^κ

i

that

ι_Xθ=









0 −α1(X) α1(X) 0

...

0 −α_λκ

i/2(X) α_λκ

i/2(X) 0







 .

Since (g⊕RR)_B^κ

i ⊂(g⊕t)_B^κ

i, we obtain for every X ∈(g⊕RR)_B^κ

i

Pf(ι_Xθ) = 1 (−2π)^λκi/2

λ^κ_i/2

Y

j=1

α_j(X). (4.5)

Let k be a complement of (g⊕RR)_B^κ

i in g⊕RR. Then, by the definition of the Cartan complex, we have C_g⊕RR(B_i^κ) =S((g⊕RR)^∗_Bκ

i)⊗C_k(B_i^κ), d_g⊕RR= 1⊗d_k, and H_g⊕RR(B_i^κ) = S((g⊕RR)^∗_Bκ

i)⊗H_k(B_i^κ). S((g⊕RR)^∗_Bκ

i) is a polynomial ring, so anyω₀ ∈S((g⊕RR)^∗_Bκ

i) with ω₀ 6= 0 is not a zero divisor inH_g⊕RR(B_i^κ). More generally, if there is an ω₀ ∈ S((g⊕RR)^∗_Bκ

i) such that ω ∈ H_g⊕RR(B_i^κ) is of the form

ω=ω₀⊗1 +terms of positive degree in H_k(B_i^κ),

then ω is not a zero divisor in Hg⊕RR(B_i^κ) (cf. also [AB83, p. 605]). Hence, for E_i^κ not to be a zero divisor, it suffices to show that its purely polynomial part in S((g⊕RR)^∗_B^κ

i)⊗1 is not a zero divisor. Since E_i^κ is a form of degree λ^κ_i, as is Qλ^κ_i/2

j=1 αj, it follows with Equation (4.5) that E_i^κ = 1

(2π)^λκi/2

λ^κ_i/2

Y

j=1

α_j ⊗1 +terms of positive degree in H_k(B_i^κ),

i.e., it suffices to show that Qλ^κ_i/2

j=1 α_j 6≡ 0 on (g⊕RR)_B^κ

i. Suppose thatQλ^κ_i/2 j=1 α_j vanishes on (g⊕RR)_B^κ

i. Then, since the linear forms α_j either vanish on (g⊕ RR)_B^κ

i or have a kernel of codimension 1, there existed an α_j0 that vanished on all of (g⊕RR)_B^κ

i. By (4.4), this meant that a two-dimensional subspace ofν⁻B_i^κ vanished under (g⊕RR)_B^κ

i. This, however, contradicts Lemma 4.1.2.

Recalling Propositions 3.3.3 and 3.4.6, we also set E_i^κ =e_G(ν⁻B_i^κ,F) by abuse of notation. We obtain an injective map

⊕(·E_i^κ) : M

i

H_G^∗−λκi(B_i^κ,F)−→M

i

H_G^∗(B^κ_i,F). (4.6) Now, set Y_s^± := {±Ψ^Xs+1|_Ys ≥ 0}. We then have Y_s+1 = (Ψ^Xs+1|_Ys)⁻¹(0) = Y_s⁺∩Y_s⁻. Let 0 < κ₀ < κ₁ < ... < κ_m be the critical values of Ψ^Xs+1|_Ys attained on Y_s⁺. Consider the following diagram, in which the top row is the long exact sequence of the pair((Y_s⁺)^κj+j,(Y_s⁺)^κj−j), see Proposition 3.3.12, and the vertical arrow on the right and the diagonal arrow are the restriction to ∪B_i^κj.

... //H_G^∗(Ys^κj+j, Ys^κj−j,F)^{hj //}

∼=

H_G^∗((Y_s⁺)^κj+j,F) ^//

H_G^∗((Y_s⁺)^κj−j,F) ^//...

L

iH_G,c^∗ (ν⁻B_i^κj,F)

**

∼=

L

iH^∗−λ

κj i

G (B_i^κj,F)

⊕(·E_i^κj)

//L

iH_G^∗(B_i^κj,F).

The following argument is similar to that in [GT10, Theorem 7.1]. The Isomor-phisms (4.2) and (4.3) yield that the two vertical arrows on the left are isomor-phisms. Note that H_G^∗(Ys^κj+j, Ys^κj−j,F) =H_G^∗((Y_s⁺)^κj+j,(Y_s⁺)^κj−j,F) by exci-sion and homotopy equivalence. The upper part of the diagram commutes, because under the isomorphism ϕ : H_G(Ys^κj+j, Ys^κj−j,F) ∼= ⊕_iH_G(ν⁻B_i^κj,F), the sub-manifoldsB_i^κj ⊂Ys^κj+j are preserved, that is, mapped to the zero section ofν⁻B_i^κj. Hence, ⊕|_Bκj

i ◦ϕ=⊕|_Bκj

i ◦h_j, where h_j :H_G(Ys^κj+j, Ys^κj−j,F)→ H_G(Ys^κj+,F) denotes the projection onto the first factor. It follows that the composition of h_j and the right vertical arrow is the restriction toB_i^κj of the first factor. By Theorem 3.4.7, the remainder of the diagram is commutative. Multiplication by ⊕(·E_i^κ) is injective by (4.6), therefore h_j has to be injective.

The injectivity ofh_j yields that the long exact sequence turns into the short exact sequences

0→H_G^∗(Y_s^κj+j, Y_s^κj−j,F)→H_G^∗((Y_s⁺)^κj+j,F)−→^ιj H_G^∗((Y_s⁺)^κj−j,F)→0, hence, the natural map ιj is surjective. Furthermore, we know that the homo-topy type does not change before crossing a critical value by Theorem A.2, thus H_G^∗((Y_s⁺)^κj−j,F) = H_G^∗((Y_s⁺)^{κj−1+j−1},F). In particular, H_G^∗((Y_s⁺)^κ0−0,F) = H_G^∗(Ys+1,F) and H_G^∗((Ys)⁺,F) = H_G^∗((Y_s⁺)^κm+m,F). This yields the following sequence of surjective maps

H_G^∗((Ys)⁺,F) = H_G^∗((Y_s⁺)^κm+m,F)→ · · · →H_G^∗((Y_s⁺)^κ0+0,F)→H_G^∗(Ys+1,F).

Thus, the natural map H_G^∗(Y_s⁺,F)→H_G^∗(Y_s+1,F) is surjective.

Analogous reasoning with −Ψ^Xs+1|_Ys and, hence, the Euler classes of the positive normal bundles yields the surjectivity of H_G^∗(Y_s⁻,F)→H_G^∗(Y_s+1,F).

We consider the Mayer-Vietoris sequence (see Proposition 3.3.6) of the two open sets {x ∈Y_s | ±Ψ^Xs+1(x) >−δ} ⊂ Y_s. For sufficiently small δ >0, these sets are G×T-homotopy equivalent toY_s^±. The epimorphismsH_G^∗(Y_s^±,F)→H_G^∗(Y_s+1,F) turn the Mayer-Vietoris sequence into the short exact sequences

0→H_G^∗(Ys,F)^(j

+)^∗⊕(j⁻)^∗

→ H_G^∗(Y_s⁺,F)⊕H_G^∗(Y_s⁻,F)⁽ⁱ

+)^∗−(i⁻)^∗

→ H_G^∗(Ys+1,F)→0, (4.7) where j^±: Y_s^± ,→ Ys and i^±: Ys+1 ,→ Y_s^± denote the inclusions. We claim that the composition of these maps induces an epimorphism in equivariant basic co-homology. So let ω ∈ H_G^∗(Ys+1,F) be arbitrary. We know that (i^±)^∗ are sur-jective, hence there exist η^± ∈ H_G^∗(Y_s^±,F) such that (i^±)^∗(η^±) = ω. But this means that η⁺ +η⁻ ∈ ker((i⁺)^∗ −(i⁻)^∗) = im((j⁺)^∗ + (j⁻)^∗), i.e., there exists σ ∈ H_G^∗(Ys,F) such that η⁺ +η⁻ = (j⁺)^∗(σ) + (j⁻)^∗(σ). This, however, yields ω= (i⁺)^∗◦(j⁺)^∗(σ) = (i⁻)^∗◦(j⁻)^∗(σ) and concludes the proof of the surjectivity

H_G^∗(Ys,F)H_G^∗(Ys+1,F).

Iteration fors= 0, ..., r−1yields the desired sequence of epimorphisms H_G^∗(M,F) =H_G^∗(Y₀,F)→H_G^∗(Y₁,F)→...→H_G^∗(Y_r,F) =H_G^∗(Ψ⁻¹(0),F).

Remark 4.1.4. The idea to obtain the Kirwan map as the composition of surjec-tive maps H_G^∗(Y_s,F) H_G^∗(Y_s+1,F) stems from the approach used in [GGK02, Proof of Theorem G.13] and [BL10, Proof of Theorem 3.4]. To obtain surjectivity, Euler class arguments were also used in [BL10].

Im Dokument Torus Actions on K-Contact Manifolds : Basic Kirwan Surjectivity, Localization, and Residue Formula (Seite 55-67)