The totally nonnegative Grassmannian is a ball

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.


Introduction
The prototypical example of a closed ball of interest in topological combinatorics is a convex polytope. Over the past few decades, an analogy between convex polytopes, and certain spaces appearing in total positivity and in electrical resistor networks, has emerged [Lus94,FS00,Pos07,CdVGV96,CIM98]. One motivation for this analogy is that these latter spaces come equipped with cell decompositions whose face posets share a number of common features with the face posets of polytopes [Wil07,Her14,RW10]. A new motivation for this analogy comes from recent developments in high-energy physics, where physical significance is ascribed to certain differential forms on positive spaces which generalize convex polytopes [AHBC + 16, AHT14,AHBL17]. In this paper we show in several fundamental cases that this analogy holds at the topological level: the spaces themselves are closed balls.
1.1. The totally nonnegative Grassmannian. Let Gr(k, n) denote the Grassmannian of k-planes in R n . Postnikov [Pos07] defined its totally nonnegative part Gr ≥0 (k, n) as the set of X ∈ Gr(k, n) whose Plücker coordinates are all nonnegative. The totally nonnegative Grassmannian is not a polytope, but Postnikov conjectured that it is the 'next best thing', namely, a regular CW complex homeomorphic to a closed ball. He found a cell decomposition of Gr ≥0 (k, n), where each open cell is specified by requiring some subset of the Plücker coordinates to be strictly positive, and requiring the rest to equal zero.
Over the past decade, much work has been done towards Postnikov's conjecture. The face poset of the cell decomposition (described in [Rie99,Rie06,Pos07]) was shown to be shellable by Williams [Wil07]. Postnikov, Speyer, and Williams [PSW09] showed that the cell decomposition is a CW complex, and Rietsch and Williams [RW10] showed that it is This brings the analogy between totally nonnegative spaces and polytopes beyond the level of face posets.
Let k, m, n be nonnegative integers with k + m ≤ n, and Z be a (k + m) × n matrix whose (k + m) × (k + m) minors are all positive. We regard Z as a linear map R n → R k+m , which induces a map Z Gr on Gr(k, n) taking the subspace X to the subspace {Z(v) : v ∈ X}. The (tree) amplituhedron A n,k,m (Z) is the image of Gr ≥0 (k, n) in Gr(k, k + m) under the map Z Gr [AHT14, Section 4]. When k = 1, the totally nonnegative Grassmannian Gr ≥0 (1, n) is a simplex in P n−1 , and the amplituhedron A n,1,m (Z) is a cyclic polytope in P m [Stu88]. Understanding the topology of amplituhedra, and more generally of Grassmann polytopes [Lam16] (obtained by relaxing the positivity condition on Z), was one of the main motivations of our work.
We now take m to be even, and Z = Z 0 such that the rows of Z 0 span the unique element of Gr ≥0 (k + m, n) invariant under the Z/nZ-cyclic action (cf. [Kar19]). We call A n,k,m (Z 0 ) the cyclically symmetric amplituhedron. When k = 1 and m = 2, A n,1,2 (Z 0 ) is a regular n-gon in the plane. More generally, A n,1,m (Z 0 ) is a polytope whose vertices are n regularly spaced points on the trigonometric moment curve in P m .
Theorem 1.2. The cyclically symmetric amplituhedron A n,k,m (Z 0 ) is homeomorphic to a km-dimensional closed ball.
It is expected that every amplituhedron is homeomorphic to a closed ball. The topology of amplituhedra and Grassmann polytopes is not well understood in general; see [KW19,AHTT18] for recent work.
1.4. The compactification of the space of planar electrical networks. Let Γ be an electrical network consisting only of resistors, modeled as an undirected graph whose edge weights (conductances) are positive real numbers. The electrical properties of Γ are encoded by the response matrix Λ(Γ) : R n → R n , sending a vector of voltages at n distinguished boundary vertices to the vector of currents induced at the same vertices. The response matrix can be computed using (only) Kirchhoff's law and Ohm's law. Following Curtis, Ingerman, and Morrow [CIM98] and Colin de Verdière, Gitler, and Vertigan [CdVGV96], we consider the space Ω n of response matrices of planar electrical networks: those Γ embedded into a disk, with boundary vertices on the boundary of the disk. This space is not compact; a compactification E n was defined by the third author in [Lam18]. It comes equipped with a natural embedding ι : E n → Gr ≥0 (n − 1, 2n). We exploit this embedding to establish the following result.
Theorem 1.3. The space E n is homeomorphic to an n 2 -dimensional closed ball. A cell decomposition of E n was defined in [Lam18], extending earlier work in [CIM98,CdVGV96]. The face poset of this cell decomposition had been defined and studied by Kenyon [Ken12, Section 4.5.2]. Theorem 1.3 says that the closure of the unique cell of top dimension in E n is homeomorphic to a closed ball. In [Lam15], the third author showed that the face poset of the cell decomposition of E n is Eulerian, and conjectured that it is shellable. Hersh and Kenyon recently proved this conjecture [HK21]. Björner's results [Bjö84] therefore imply that this poset is the face poset of some regular CW complex homeomorphic to a ball.
We expect that E n forms such a CW complex, so that the closure of every cell of E n is homeomorphic to a closed ball. Proving this remains an open problem.
1.5. Outline. In Section 2, we prove a topological lemma (Lemma 2.3) which essentially states that if one can find a contractive flow (defined below) on a submanifold of R N , then its closure is homeomorphic to a ball. We use Lemma 2.3 in Sections 3 to 6 to show that the four spaces discussed in Sections 1.1 to 1.4 are homeomorphic to closed balls. To do so, in each case we consider a natural flow on the underlying space, and show that it satisfies the contractive property by introducing novel coordinates on the space.
Acknowledgements. We thank Patricia Hersh and Lauren Williams for helpful comments, and anonymous referees for many suggestions leading to improvements in the exposition.

Contractive flows
In this section we prove Lemma 2.3, which we will repeatedly use in establishing our main theorems. Consider a real normed vector space (R N , · ). Thus for each r > 0, the closed ball B N r := {p ∈ R N : p ≤ r} of radius r is a compact convex body in R N whose interior contains the origin. We denote its boundary by ∂B N r , which is the sphere of radius r. Definition 2.1. We say that a map f : R × R N → R N is a contractive flow if the following conditions are satisfied: (1) the map f is continuous; (2) for all p ∈ R N and t 1 , t 2 ∈ R, we have f (0, p) = p and f (t 1 + t 2 , p) = f (t 1 , f (t 2 , p)); and (3) for all p = 0 and t > 0, we have f (t, p) < p .
The condition (2) says that f induces a group action of (R, +) on R N . In particular, f (t, p) = q is equivalent to f (−t, q) = p, so (3) implies that if t = 0 and f (t, p) = p, then p = 0. The converse to this statement is given below in Lemma 2.2(i).
(ii) This follows from (3) and the fact that f induces a group action of R on R N , once we know that f (t, p) is never 0. But if f (t, p) = 0 then f (−t, 0) = p, which contradicts part (i).
Lemma 2.3. Let Q ⊂ R N be a smooth embedded submanifold of dimension d ≤ N , and f : R × R N → R N a contractive flow. Suppose that Q is bounded and satisfies the condition Then the closure Q is homeomorphic to a closed ball of dimension d, and Q \ Q is homeomorphic to a sphere of dimension d − 1.
Note that any open subset of R N is a smooth embedded submanifold of dimension N .
Proof. Since Q is bounded, its closure Q is compact. By Lemma 2.2(iii) and (2.4) we have 0 ∈ Q, and therefore 0 ∈ Q. Because Q is smoothly embedded, we can take r > 0 sufficiently small so that B := B N r ∩ Q is homeomorphic to a closed ball of dimension d. We let ∂B denote (∂B N r ) ∩ Q, which is a (d − 1)-dimensional sphere. For any p ∈ R N \ {0}, consider the curve t → f (t, p) starting at p and defined for all t ∈ R. By Lemma 2.2(ii), this curve intersects the sphere ∂B N r for a unique t ∈ R, which we denote by t r (p). Also, for p ∈ Q \ {0}, define t ∂ (p) ∈ (−∞, 0] as follows. Let T (p) := {t ∈ R : f (t, p) ∈ Q}. We have 0 ∈ T (p), and T (p) is bounded from below by Claim. The functions t r and t ∂ are continuous on Q \ {0}.
Proof of Claim. First we prove that t r is continuous on R N \ {0}. It suffices to show that the preimage of any open interval I ⊂ R is open. To this end, let q ∈ t −1 r (I). Take t 1 , t 2 ∈ I with t 1 < t r (q) < t 2 . By Lemma 2.2(ii), we have f (t 1 , q) > r > f (t 2 , q) . Note that the map γ 1 : is an open neighborhood of q, whose image under t r is contained in (t 1 , t 2 ) ⊂ I. This shows that t r is continuous on R N \ {0}.
Next, let us define We now prove that the map t ∂ : Q \ {0} → R is continuous, by a very similar argument. Let I ⊂ R be an open interval and consider a point q ∈ t −1 ∂ (I). Take t 1 , t 2 ∈ I with This finishes the proof of the claim.
Define the maps α : Q → B and β : B → Q by for p = 0, and α(0) := 0, β(0) := 0. Let us verify that α sends Q inside B and β sends Now we check that α and β are inverse maps. For any p ∈ Q and ∆t ∈ R such that f (∆t, p) ∈ Q, we have We can similarly verify that β(α(p)) = p, by instead taking ∆t := t r (p) − t ∂ (p).
By the claim, t r and t ∂ are continuous on Q \ {0}, so α is continuous everywhere except possibly at 0. Also, t r (p) > t ∂ (p) for all p ∈ Q\{0}, so α is continuous at 0 by Lemma 2.2(ii). Thus α is a continuous bijection from a compact space to a Hausdorff space, so it is a homeomorphism. This shows that Q is homeomorphic to a closed d-dimensional ball.
It remains to prove that Q \ Q is homeomorphic to a (d − 1)-dimensional sphere. We claim that α restricts to a homeomorphism from Q \ Q to ∂B. We need to check that α sends Q \ Q inside ∂B, and β sends ∂B inside Q \ Q. To this end, let p ∈ Q \ Q. By condition (2), we

The totally nonnegative Grassmannian
Let Gr(k, n) denote the real Grassmannian, the space of all k-dimensional subspaces of R n . We set [n] := {1, . . . , n}, and let [n] k denote the set of k-element subsets of [n]. For X ∈ Gr(k, n), we denote by (∆ I (X)) I∈( [n] k ) ∈ P ( n k )−1 the Plücker coordinates of X: ∆ I (X) is the k × k minor of X (viewed as a k × n matrix modulo row operations) with column set I.
Recall that Gr ≥0 (k, n) is the subset of Gr(k, n) where all Plücker coordinates are nonnegative (up to a common scalar). We also define the totally positive Grassmannian Gr >0 (k, n) as the subset of Gr ≥0 (k, n) where all Plücker coordinates are positive.
3.1. Global coordinates for Gr ≥0 (k, n). For each k and n, we introduce several distinguished linear operators on R n . Define the left cyclic shift S ∈ gl n (R) = End(R n ) by S(v 1 , . . . , v n ) := (v 2 , . . . , v n , (−1) k−1 v 1 ). The sign (−1) k−1 can be explained as follows: if we pretend that S is an element of GL n (R), then the action of S on Gr(k, n) preserves Gr ≥0 (k, n) (it acts on Plücker coordinates by rotating the index set [n]).
Note that the transpose S T of S is the right cyclic shift given by We endow R n with the standard inner product, so that τ (being symmetric) has an orthogonal basis of eigenvectors u 1 , . . . , u n ∈ R n corresponding to real eigenvalues λ 1 ≥ · · · ≥ λ n . Let X 0 ∈ Gr(k, n) be the linear span of u 1 , . . . , u k . The following lemma implies that X 0 is totally positive and does not depend on the choice of eigenvectors u 1 , . . . , u n .
(i) The eigenvalues λ 1 ≥ · · · ≥ λ n are given as follows, depending on the parity of k: The Plücker coordinates of X 0 are given by For an example in the case of Gr(2, 4), see Section 3.5. (We remark that in the example, the Plücker coordinates of X 0 are scaled by a factor of 2 compared to the formula above.) Proof. In this proof, we work over C. Let ζ ∈ C be an nth root of (−1) k−1 . There are n such values of ζ, each of the form ζ = e iπm/n for some integer m congruent to k − 1 modulo 2. Let z m := (1, ζ, ζ 2 , . . . , ζ n−1 ) ∈ C n . We have S(z m ) = ζz m and S T (z m ) = ζ −1 z m , so The n distinct z m 's are linearly independent (they form an n × n Vandermonde matrix with nonzero determinant), so they give a basis of C n of eigenvectors of τ .
Then the Plücker coordinates of X 0 are the k ×k minors of M (up to a common nonzero complex scalar), which can be computed explicitly by Vandermonde's identity after appropriately rescaling the columns. We refer the reader to [Kar19, Proposition 2.5] for details.
In other words, the entries of A are the usual coordinates on the big Schubert cell of Gr(k, n) with respect to the basis u 1 , . . . , u n of R n , this Schubert cell being In particular, φ is a smooth embedding, and it sends the zero matrix to X 0 . For an example in the case of Gr(2, 4), see Section 3.5.
Proof. Let X ∈ Gr ≥0 (k, n) be a totally nonnegative subspace. We need to show that X ∩ span(u k+1 , . . . , u n ) = 0. Suppose otherwise that there exists a nonzero vector v in this intersection. Extend v to a basis of X, and write this basis as the rows of a k × n matrix M . Because X is totally nonnegative, the nonzero k × k minors of M all have the same sign (and at least one minor is nonzero, since M has rank k). Also let M 0 be the k × n matrix with rows u 1 , . . . , u k . By We have shown that the restriction of φ −1 to Gr ≥0 (k, n) yields an embedding Gr ≥0 (k, n) → Mat(k, n − k) R k(n−k) whose restriction to Gr >0 (k, n) is smooth.
3.2. Flows on Gr(k, n). For g ∈ GL n (R), we let g act on Gr(k, n) by taking the subspace X to g · X := {g(v) : v ∈ X}. We let 1 ∈ GL n (R) denote the identity matrix, and for x ∈ gl n (R) we let exp(x) := ∞ j=0 x j j! ∈ GL n (R) denote the matrix exponential of x. We examine the action of exp(tS) and exp(tτ ) on Gr(k, n).
To see why this is sufficient, let X ∈ Gr ≥0 (k, n) and t > 0. By part (i), we have exp(tS)·X ∈ Gr ≥0 (k, n), so we just need to show that exp(tS) · X ∈ Gr >0 (k, n). Suppose otherwise that exp(tS) · X / ∈ Gr >0 (k, n). Then applying part (ii) to exp(tS) · X, we get that exp((t + t )S) · X / ∈ Gr ≥0 (k, n) for t < 0 sufficiently close to zero. But by part (i), we know that exp((t + t )S) · X ∈ Gr ≥0 (k, n) for all t in the interval [−t, 0]. This is a contradiction. Now we prove parts (i) and (ii). We will make use of the operator 1 + tS, which belongs to GL n (R) for |t| < 1. Note that if [M 1 | · · · | M n ] is a k × n matrix representing X, then a k × n matrix representing (1 + tS) · X is We can evaluate the k × k minors of M using multilinearity of the determinant. We obtain where i 1 + 1 , . . . , i k + k are taken modulo n. Therefore (1 + tS) · X ∈ Gr ≥0 (k, n) for X ∈ Gr ≥0 (k, n) and t ∈ [0, 1). Since exp(tS) = lim j→∞ 1 + tS j j and Gr ≥0 (k, n) is closed, we obtain exp(tS) · X ∈ Gr ≥0 (k, n) for t ≥ 0. This proves part (i).
To prove part (ii), first note that exp(tS) = 1 + tS + O(t 2 ). By (3.6), we have where the sum is over all I ∈ [n] k obtained from I by increasing exactly one element by 1 modulo n. If we can find such I and I with ∆ I (X) = 0 and ∆ I (X) > 0, then ∆ I (exp(tS) · X) < 0 for all t < 0 sufficiently close to zero, thereby proving part (ii). In order to do this, we introduce the directed graph D with vertex set [n] k , where J → J is an edge of D if and only if we can obtain J from J by increasing exactly one element by 1 modulo n. Note that for any two vertices K and K of D, there exists a directed path from K to K : • we can get from [k] to any {i 1 < · · · < i k } by shifting k to i k , k − 1 to i k−1 , etc.; • similarly, we can get from any {i 1 < · · · < i k } to {n − k + 1, n − k + 2, . . . , n}; • we can get from {n − k + 1, . . . , n} to [k] by shifting n to k, n − 1 to k − 1, etc.
k with ∆ K (X) = 0 and ∆ K (X) > 0, and consider a directed path from K to K . It goes through an edge I → I with ∆ I (X) = 0 and ∆ I (X) > 0, as desired. Now we consider exp(tτ ) = exp(t(S + S T )). Recall that S and S T are the left and right cyclic shift maps, so by symmetry Lemma 3.5 holds with S replaced by S T . Also, S and S T commute, so exp(tτ ) = exp(tS) exp(tS T ). We obtain the following.
Let us see how exp(tτ ) acts on matrices A ∈ Mat(k, n − k). Note that τ (u i ) = λ i u i for 1 ≤ i ≤ n, so exp(tτ )(u i ) = e tλ i u i . Therefore exp(tτ ) acts on the basis of φ(A) in (3.3) by exp(tτ )(u i + n−k j=1 A i,j u k+j ) = e tλ i (u i + n−k j=1 e t(λ k+j −λ i ) A i,j u k+j ) for all 1 ≤ i ≤ k. Thus exp(tτ )·φ(A) = φ(f (t, A)), where by definition f (t, A) ∈ Mat(k, n−k) is the matrix with entries (f (t, A)) i,j := e t(λ k+j −λ i ) A i,j for 1 ≤ i ≤ k and 1 ≤ j ≤ n − k. (3.9) 3.3. Proof of Theorem 1.1. Consider the map f : R × Mat(k, n − k) → Mat(k, n − k) defined by (3.9). We claim that f is a contractive flow on Mat(k, n − k) equipped with the Euclidean norm Indeed, parts (1) and (2) of Definition 2.1 hold for f . To see that part (3) holds, note that for any 1 ≤ i ≤ k and 1 ≤ j ≤ n − k with A i,j = 0, we have Let us now apply Lemma 2.3 with R N = Mat(k, n − k) and Q = φ −1 (Gr >0 (k, n)). We need to know that Gr ≥0 (k, n) is the closure of Gr >0 (k, n). This was proved by Postnikov [Pos07, Section 17]; it also follows directly from Corollary 3.8, since we can express any X ∈ Gr ≥0 (k, n) as a limit of totally positive subspaces: Therefore Q = φ −1 (Gr ≥0 (k, n)). Moreover, Gr ≥0 (k, n) is closed inside the compact space P ( n k )−1 , and is therefore also compact. So, Q is compact (and hence bounded). Finally, the property (2.4) in this case is precisely Corollary 3.8. We have verified all the hypotheses of Lemma 2.3, and conclude that Q (and also Gr ≥0 (k, n)) is homeomorphic to a k(n − k)dimensional closed ball.
3.4. Related work. Lusztig [Lus98, Section 4] used a flow similar to exp(tτ ) to show that (G/P ) ≥0 is contractible. Our flow can be thought of as an affine (or loop group) analogue of his flow, and is closely related to the whirl matrices of [LP12]. We also remark that Ayala, Kliemann, and San Martin [AKSM04] used the language of control theory to give an alternative development in type A of Lusztig's theory of total positivity. In that context, exp(tτ ) (t > 0) lies in the interior of the compression semigroup of Gr ≥0 (k, n), and X 0 is its attractor.
Marsh and Rietsch defined and studied a superpotential on the Grassmannian in the context of mirror symmetry [MR20, Section 6]. It follows from results of Rietsch [Rie08] (see [Kar19, Corollary 3.12]) that X 0 is, rather surprisingly, also the unique totally nonnegative critical point of the q = 1 specialization of the superpotential. However, the superpotential is not defined on the boundary of Gr ≥0 (k, n). The precise relationship between τ and the gradient flow of the superpotential remains mysterious. Gr(2, 4). The matrix τ = S + S T ∈ gl 4 (R) and an orthogonal basis of real eigenvectors u 1 , u 2 , u 3 , u 4 are

Example: the case
The embedding φ : Mat(2, 2) → Gr(2, 4) sends the matrix A = a b c d to Above we are identifying X ∈ Gr(2, 4) with a 2 × 4 matrix whose rows form a basis of X.
In terms of Plücker coordinates ∆ ij = ∆ {i,j} (X), the map φ is given by and its inverse is given by The point X 0 = φ(0) = span(u 1 , u 2 ) ∈ Gr >0 (2, 4) has Plücker coordinates In general, using the embedding φ we can describe Gr ≥0 (k, n) as the subset of R k(n−k) where some n k polynomials of degree at most k are nonnegative.

The totally nonnegative part of the unipotent radical of GL n (R)
Recall from Section 1.2 that U denotes the unipotent group of upper-triangular matrices in GL n (R) with 1's on the diagonal, and U ≥0 is its totally nonnegative part, where all minors are nonnegative. We also let V ⊂ U be the set of x ∈ U whose superdiagonal entries x i,i+1 sum to n − 1, and define V ≥0 := V ∩ U ≥0 . We may identify V ≥0 with the link of the identity matrix 1 in U ≥0 . In this section, we prove the following result. It is a special case of a result of Hersh [Her14], who established the corresponding result in general Lie type, and in addition for all the lower-dimensional cells in the Bruhat stratification.
Theorem 4.1 ([Her14]). The space V ≥0 is homeomorphic to an n 2 − 1 -dimensional closed ball. The space U ≥0 is homeomorphic to a closed half-space in R ( n 2 ) .
Let e ∈ gl n (R) be the upper-triangular principal nilpotent element, which has 1's on the superdiagonal and 0's elsewhere. We wish to consider the flow on V ≥0 generated by exp(te), which we remark was used by Lusztig to show that U ≥0 is contractible [Lus98, Section 4]. However, we must take care to define a flow which preserves V and not merely U . To this end, for t > 0, let ρ(t) ∈ GL n (R) be the diagonal matrix with diagonal entries (t n−1 , t n−2 , . . . , 1). Note that ρ is multiplicative, i.e., ρ(s)ρ(t) = ρ(st). Define a(t) : U → U by Lemma 4.3. The map a(·) defines an action of the multiplicative group R >0 on V , i.e., a(t) · x ∈ V, a(1) · x = x, and a(s) · (a(t) · x) = a(st) · x for all s, t > 0 and x ∈ V .
Proof. To prove part (i), we must show that exp((t − 1)e)x ∈ U >0 for x ∈ V ≥0 and t > 1. This follows by writing the relevant minors of exp((t − 1)e)x via the Cauchy-Binet identity, using the fact that exp((t − 1)e) ∈ U >0 (see [Lus94,Proposition 5.9]). For part (ii), because a(·) is multiplicative (Lemma 4.3) it suffices to prove that t → a(t) · x ∞ is decreasing at t = 1. Using the description of the entries of a(t) given by setting s = t in (4.4), we get as t → 1, where we set b i+1,j (x) := 0 if i + 1 = j. Then for any i < j with |b i,j (x)| = x ∞ , we have |b i+1,j (x)/c| < |b i,j (x)|, and so |b i,j (a(t) · x)| is decreasing at t = 1.
Let us now show that the hypotheses of Lemma 2.3 hold, with R N = W and Q = b(V >0 ). Lemma 4.6(i) implies that Q = b(V ≥0 ), and also verifies (2.4). Finally, Q is bounded because V ≥0 is bounded, e.g., one can prove by induction on j − i that for any x ∈ V ≥0 , (Alternatively, see [Lus94, proof of Proposition 4.2].) Thus Lemma 2.3 implies that Q (and hence V ≥0 ) is homeomorphic to an N -dimensional closed ball.
For U ≥0 , we use the dilation action of R >0 on U ≥0 , where t ∈ R >0 acts by multiplying all entries x i,j on the (j − i)-th diagonal (above the main diagonal) by t j−i . Therefore U ≥0 is homeomorphic to the open cone over the compact space V ≥0 . That is, U ≥0 is homeomorphic to the quotient space of R ≥0 × V ≥0 by the subspace 0 × V ≥0 , with the identity matrix 1 ∈ U ≥0 corresponding to the cone point.
which converges to exp(e) =   1 1 1/2 0 1 1 0 0 1   as t → ∞. The coordinates b i,j from (4.5) are We can then try to verify Lemma 4.6 directly in this case (this is a nontrivial exercise). ♦

The cyclically symmetric amplituhedron
Let k, m, n be nonnegative integers with k + m ≤ n and m even, and let S, τ ∈ gl n (R) be the operators from Section 3.1. Let λ 1 ≥ · · · ≥ λ n ∈ R be the eigenvalues of τ corresponding to orthogonal eigenvectors u 1 , . . . , u n . In this section, we assume that these eigenvectors have norm 1. Recall from Lemma 3.1(i) that λ k > λ k+1 . Since m is even, we have (−1) k+m−1 = (−1) k−1 and λ k+m > λ k+m+1 . Let Z 0 denote the (k + m) × n matrix whose rows are u 1 , . . . , u k+m . By Lemma 3.1(ii), the (k +m)×(k +m) minors of Z 0 are all positive (perhaps after replacing u 1 with −u 1 ). We may also think of Z 0 as a linear map R n → R k+m . Since the vectors u 1 , . . . , u n are orthonormal, this map takes u i to the ith unit vector e i ∈ R k+m if i ≤ k + m, and to 0 if i > k + m. Recall from Section 1.3 that Z 0 induces a map (Z 0 ) Gr : Gr ≥0 (k, n) → Gr(k, k + m), whose image is the cyclically symmetric amplituhedron A n,k,m (Z 0 ). We remark that if g ∈ GL k+m (R), then A n,k,m (gZ 0 ) and A n,k,m (Z 0 ) are related by the automorphism g of Gr(k, k + m), so the topology of A n,k,m (Z 0 ) depends only on the row span of Z 0 in Gr(k + m, n).
Proof of Theorem 1.2. We consider the map φ : Mat(k, n − k) → Gr(k, n) defined in (3.3). We write each k×(n−k) matrix A ∈ Mat(k, n−k) as [A | A ], where A and A are the k×m and k × (n − k − m) submatrices of A with column sets {1, . . . , m} and {m + 1, . . . , n − k}, respectively. We introduce a projection map We claim that there exists an embedding γ : A n,k,m (Z 0 ) → Mat(k, m) making the following diagram commute: Mat(k, m) be a matrix such that φ(A) ∈ Gr ≥0 (k, n). Then the element (Z 0 ) Gr (φ(A)) of Gr(k, k + m) is the row span of the k × (k + m) matrix [Id k | A ], where Id k denotes the k × k identity matrix. Thus A n,k,m (Z 0 ) = (Z 0 ) Gr (Gr ≥0 (k, n)) lies inside the Schubert cell Every element Y of this Schubert cell is the row span of [Id k | A ] for a unique A , and we define γ(Y ) := A . Thus γ embeds A n,k,m (Z 0 ) inside Mat(k, m), and (5.1) commutes. Now we define Q := π(φ −1 (Gr >0 (k, n))) ⊂ Mat(k, m).
We know from Section 3.3 that φ −1 (Gr >0 (k, n)) is an open subset of Mat(k, n) whose closure φ −1 (Gr ≥0 (k, n)) is compact. Note that π is an open map (since it is essentially a projection R k(n−k) → R km ), so Q is an open subset of Mat(k, m). The closure Q = π(φ −1 (Gr ≥0 (k, n))) of Q is compact. By (5.1), Q is homeomorphic to A n,k,m (Z 0 ).
Let f : R × Mat(k, n − k) → Mat(k, n − k) be the map defined by (3.9), and define a similar map f 0 : R × Mat(k, m) → Mat(k, m) by That is, f 0 (t, π(A)) = π(f (t, A)) for all t ∈ R and A ∈ Mat(k, n − k). We showed in Section 3.3 that f is a contractive flow, so f 0 is also a contractive flow. We also showed that f (t, φ −1 (Gr ≥0 (k, n))) ⊂ φ −1 (Gr >0 (k, n)) for t > 0, and applying π to both sides shows that Thus Lemma 2.3 applies to Q and f 0 , showing that Q (and hence A n,k,m (Z 0 )) is homeomorphic to a km-dimensional closed ball.
Note that this τ differs in the top-right and bottom-left entries from the one in Section 3.5, because k is odd rather than even. Also, here the eigenvectors are required to have norm 1.
This line gets sent by (Z 0 ) Gr to the row span of the matrix v · Z T 0 = 1 a b . Finally, γ sends this element of Gr(1, 3) to the matrix a b , so (5.1) indeed commutes.
In order for φ(A) to land in Gr ≥0 (1, 4), the coordinates of v must all have the same sign, and since their sum is 2, they must all be nonnegative: These linear inequalities define a tetrahedron in R 3 Mat(1, 3) with the four vertices 0, ± √ 2, −1 , ± √ 2, 0, 1 . The projection π = γ • (Z 0 ) Gr • φ sends this tetrahedron to a square in R 2 Mat(1, 2) with vertices 0, ± √ 2 , ± √ 2, 0 . This square is a km-dimensional ball, as implied by Theorem 1.2. We note that when k = 1, the amplituhedron A n,k,m (Z) (for any (k +m)×n matrix Z with positive maximal minors) is a cyclic polytope in the projective space Gr(1, m + 1) = P m [Stu88], and is therefore homeomorphic to a km-dimensional closed ball. The case of k ≥ 2 and Z = Z 0 remains open. ♦ 6. The compactification of the space of electrical networks 6.1. A slice of the totally nonnegative Grassmannian. We recall some background on electrical networks, and refer the reader to [Lam18] and Example 6.4 for details. Let R ( 2n n−1 ) have basis vectors e I for I ∈ [2n] n−1 , and let P ( 2n n−1 )−1 denote the corresponding projective space. We define Let N C n denote the collection of non-crossing partitions of [2n] odd , i.e., set partitions of [2n] odd such that there do not exist i < j < i < j in [2n] odd and distinct parts I and J with i, i ∈ I and j, j ∈ J. Each σ ∈ N C n comes with a dual non-crossing partition (or Kreweras complement)σ of [2n] even , defined to be the coarsest non-crossing partition of [2n] even such that σ ∪σ is a non-crossing partition of [2n]. We call a subset I ∈ [2n] n−1 concordant with σ if every part of σ and every part ofσ contains exactly one element not in I. Let A σ ∈ R ( 2n n−1 ) be the sum of e I over all I concordant with σ, and let H be the linear subspace of P ( 2n n−1 )−1 spanned by the images of A σ for σ ∈ N C n .