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Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections SMOOTHING MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS THOMAS BAIRD AND DANIEL A. RAMRAS n Abstract. Let X ⊂ R be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M ! X are homotopic (in X) to C1 maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds. 1. Introduction The first goal of this paper is to prove the following result about the differential topology of algebraic sets. Theorem 1.1 (Section2) . Let X ⊂ Rn be a (possibly singular) real algebraic set, and let f : M ! X be a continuous map from a smooth, closed manifold M. Then there exists a map g : M ! X, and a homotopy H : M × I ! X connecting f and g g, such that the composite M ! X,! Rn is C1. The problem of smoothing maps into algebraic sets seems natural, but we have not found mention of it in the literature. We consulted several experts in real algebraic geometry; some expected our result to hold, and some did not. Our proof proceeds by embedding X as the singular set of an irreducible, quasi- projective variety Y and using a resolution of singularities Ye ! Y for which the inverse image of X is a divisor with normal crossing singularities. Basic facts about neighborhoods of algebraic sets then reduce the problem to the case of normal crossing divisors, which can be handled by differential-geometric means. We note that a small modification of our arguments can prove a version of this result for smoothing out maps from non-compact manifolds into compact algebraic sets. In the remainder of the paper, we apply Theorem 1.1 to study the topology of spaces of flat connections over aspherical manifolds and their associated moduli spaces. In particular, given a discrete group Γ such that BΓ has the homotopy type of a finite CW complex, we study the functorial map B (1) Hom(Γ; GLn(C)) −! Map∗(BΓ; BGLn(C)); from the space of complex-valued group homomorphisms to the space of based continuous maps between classifying spaces. If BΓ is homotopy equivalent to a smooth manifold, then (1) may be interpreted as the inclusion of the moduli space of based flat connections into the moduli space of based connections. We prove The first author was partly supported by an NSERC discovery grant. The second author was partially supported by NSF grants DMS-0804553 and DMS-0968766. 2010 Mathematics Subject Classification. Primary 14P05, 53C05; Secondary 57R20, 55R37. 1 2 THOMAS BAIRD AND DANIEL A. RAMRAS Theorem 1.2 (Section4) . Let Γ be a discrete group such that BΓ has the homotopy k type of a finite, d{dimensional CW complex and let βk(BΓ) = Rank(H (BΓ; Q)). Then for each ρ 2 Hom(Γ; GLn(C)) and each m > 0, the induced map on homotopy groups B∗ : πm (Hom(Γ; GLn(C)); ρ) ! πm (Map∗(BΓ;BGLn(C)); Bρ) P1 satisfies Rank(coker(B∗)) > i=1 βm+2i(BΓ) so long as n > (m + d)=2. The strategy behind the proof of Theorem 1.2 is as follows. The space BGLn(C) n classifies C -vector bundles, so an element in πm (Map∗(BΓ;BGLn(C)); Bρ) deter- mines a vector bundle (up to isomorphism) over Sm ×BΓ. Those classes lying in the m image of B∗ correspond to vector bundles over S × BΓ which admit a connection that is “flat in the BΓ direction." We then deduce, using Chern-Weil theory, that certain Chern classes of these bundles must vanish (see Theorem 3.5). Theorem 1.1 is necessary to make this argument rigorous: to construct the desired connection on the bundle associated to B∗([ ]), we must smooth out the underlying map m from S into the (usually singular) algebraic set Hom(Γ; GLn(C)). In Section 5.2, we prove the following consequence of Theorem 1.2 (recall that U(n) is the maximal compact subgroup of GLn(C)). Corollary 1.3 (Section 5.2). Let M be a closed, smooth, aspherical manifold of dimension d, and let E ! M be a flat principal U(n){bundle over M. Let Aflat(E) denote the space of flat, unitary connections on E. Then for each A0 2 Aflat(E) and for each 0 < m 6 2n − d − 1, we have 1 X Rank (πm(Aflat(E);A0)) > βm+2i+1(M): i=1 P1 Additionally, if i=1 β2i+1(M) > 0 and 2n − d − 1 > 0, then Aflat(E) has infinitely many path components. In particular, the space of flat connections on a flat U(n){bundle (n > 2) over an orientable aspherical 3{manifold has infinitely many path components. This result appears to be new, even for the trivial bundle on the 3{torus, and is in contrast with the case of bundles over surfaces. There, Yang{Mills theory shows that Aflat(E) is highly connected with respect to the rank of E (see Ramras [17, Proposition 4.9], for instance). One fundamental difficulty in proving Corollary 1.3 is that the space of flat connections is not smooth (even in an infinite-dimensional sense) and hence it is unclear whether paths in this space are always homotopic to smooth paths. The methods developed in this paper (Theorem 1.2 in particular) also have applications to the study of Quillen{Lichtenbaum phenomena in the stable topology of representation spaces, as observed for products of surface groups in Ramras [18]. In fact, the desire to understand these phenomena gave rise to the present work. These topics will be discussed in detail elsewhere. Organization: In Section2, we establish our results about maps into real algebraic sets, which are applied to study the characteristic classes in Section3. Theorem 1.2 is proven in Section4, and spaces of flat connections are studied in Section5. Acknowledgements The first author thanks Juan Souto and Frances Kirwan for helpful conversations. The second author thanks Ben Wieland. In addition, the suggestions of the anonymous referees helped to improve the exposition. MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS 3 2. Smoothing maps into real algebraic sets The goal of this section is to prove the following result. Theorem 2.1. Let i : X,! Rn be a real algebraic set and let M be a smooth manifold. Assume that either X or M is compact. Then for every continuous map φ : M ! X there exists a map φ0 : M ! X homotopic to φ such that i ◦ φ0 is smooth.1 The following example helps illustrate the subtlety of Theorem 2.1. Consider the subspace X ⊂ R2 defined as the union of the graph f(x; y) j y = x sin(1=x); x 2 [−1/π; 0) [ (0; 1/π]g [ f(0; 0)g with the polygonal path traveling from (−1/π; 0) to (−1/π; 1) to (1/π; 1) to (1/π; 0). 1 ∼ The subspace X is homeomorphic to S , so π1(X) = Z. However, the non-trivial 1 elements of π1(X) cannot be represented by smooth maps φ : S ! X because the curve x sin(1=x) has infinite arc length. Note, moreover, that just as with the real algebraic sets considered in Theorem 2.1, it is possible to triangulate R2 with X as a subcomplex. This follows from the Schoenflies Theorem (see, for instance, Moise [15, Chapter 10, Theorem 4]) but can in fact be done quite explicitly. For the proof of Theorem 2.1, we assume that M is compact, which is the only case needed in the paper; the argument is similar when X is compact but M is not. Lemma 2.2. Let X ⊂ Rn be a real algebraic set. There exists an irreducible real algebraic set Y ⊂ Rn+2 such that Y is homeomorphic to Rn+1 and the projection map π : Rn+2 = Rn ⊕ R2 ! Rn restricts to a homeomorphism ∼ π : Sing(Y ) −!= X: Proof. We may suppose that X is equal to the zero set of a single polynomial p = p(x1; : : : ; xn), that is: X = Z(p): 2 Introduce new variables y1; y2 and consider the polynomial g(x; y) = p(x) + 2 3 y1 + y2. Define Y = Z(g): It is not hard to verify that the corresponding inclusion Rn ,! Rn+2 identifies X bijectively with the singular locus of Y , so the projection onto Rn identifies Sing(Y ) =∼ X. 2 2 1=3 Given x1; : : : ; xn; y1 2 R, the formula y2 = (g(x; y) − p(x) − y1) gives the unique solution to g(x; y) = 0. Hence Y is homeomorphic to Rn+1. Furthermore, 2 3 the polynomial g(x; y) is irreducible, since its reduction y1 + y2 is irreducible (see, for example, Dummit and Foote [1, Section 9.1]). In particular, a map into X is smooth as a map into Rn if and only if the corresponding map into Sing(Y ) is smooth as a map into Rn+2. We identify X with Sing(Y ) in what follows. The following version of Hironaka's resolution of singularities theorem appears in Koll´ar[13, Theorem 3.27]. 1We use Euclidean topology throughout, except that \irreducible algebraic set" means irre- ducible in the Zariski topology. We will identify all real algebraic sets with their real points throughout this section. 4 THOMAS BAIRD AND DANIEL A. RAMRAS Theorem 2.3. Given a quasiprojective real algebraic variety Y ⊂ Rn+2 with sin- gular set X, there exists a nonsingular real algebraic variety Ye and a birational and projective morphism Π: Ye ! Y which restricts to an isomorphism over Y − X and for which Xe := Π−1(X) is a divisor with simple normal crossings.
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