HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER REAL SURFACES 7 1 MICHAELWEST 0 2 Abstract. WeanalysethehomotopytypesofgaugegroupsofprincipalU(n)- n bundlesassociatedtopseudoRealvectorbundlesinthesenseofAtiyah[Ati66]. a We providesatisfactory homotopy decompositions of these gauge groups into J factors in which the homotopy groups are well known. Therefore, we sub- 2 stantially build upon the low dimensional homotopy groups as provided in [BHH10]. ] T A . h Contents t a m 1. Introduction 1 Acknowledgements 2 [ 1.1. Definitions 2 1 1.2. Main Results for Real Bundles 4 v 1.3. Main Results for Quaternionic Bundles 6 0 2. Proofs of Statements 8 3 4 2.1. Real Surfaces as Z2-complexes 9 0 2.2. Equivalent Components of Mapping Spaces 11 0 2.3. Pointed Gauge Groups 13 . 1 2.4. Unpointed Gauge Groups 28 0 2.5. The Quaternionic Case 35 7 3. Tables of Homotopy Groups 39 1 References 42 : v i X r a 1. Introduction Recently, the topology of gauge groups over Real surfaces has received wide- spread interest due to their intimate ties with the moduli spaces of stable vector bundles (see [BHH10] and [Sch11]). Indeed, there have been explicit calculations of some of the topological invariants of these gauge groups. For instance, Real vector bundles over Real surfaces were originally classified in [BHH10] but more recently in [GZ15]. Cohomology calculations of the classifying spaces appeared in [LS13],[Bai14] and [Bai15]. Furthermore, some of the low dimensional homotopy groups were presented in [BHH10]. The purpose of this paper is to extend the calculations of these homotopy groups by providing homotopy decompositions of the gauge groups into products of known factors. In the coming section, we define our objects of interest and state their classi- fication results. We go on to state the results of this paper, and then proofs are 1 2 MICHAELWEST provided in Section 2. In Section 3, we present tables of homotopy groups, and compare them to those provided in [BHH10] in which we highlight a discrepancy. Acknowledgements. I would like to thank Prof. Tom Baird and the referee of this paper for suggesting the proofs of Propositions 1.9 and 1.10. Additionally, I further thank Prof. Baird for a suggested reformulation of Theorem 1.15 and for providing interesting conversations surrounding the topics in this paper. I would also like to thank Prof. Stephen Theriault for suggesting this project, along with his continued support and guidance. 1.1. Definitions. The pair (X,σ), where X is a compactconnectedRiemannsur- face, and σ is an antiholomorphic involution, will be called a Real surface. To a Real surface (X,σ) we associate the following triple (g(X),r(X),a(X)) where g(X) is the genus of X; • r(X) is number of path components of the fixed set Xσ; • a(X)=0 if X/σ is orientable and a(X)=1 otherwise. • We note that the path components of Xσ are each homeomorphic to S1. The following classification of Real surfaces was studied in [Wei83]. Theorem 1.1 (Weichold). Let (X,σ) and (X′,σ′) be Real surfaces then there is a isomorphism X X′ (in the category of Real surfaces) if and only if → ′ ′ ′ (g(X),r(X),a(X))=(g(X ),r(X ),a(X )). Furthermore, if a triple (g,r,a) satisfies one of the following conditions (1) if a=0, then 1 r g+1 and r (g+1) mod 2; ≤ ≤ ≡ (2) if a=1, then 0 r g; ≤ ≤ then there is a Real surface (X,σ) such that (g,r,a)=(g(X),r(X),a(X)). (cid:3) ThereforeaRealsurface(X,σ)iscompletelydeterminedbyitstriple(g,r,a)which we call the type of the Real surface. Let π: P X be a principal U(n)-bundle overthe underlying Riemann surface → X of the Real surface (X,σ). A lift of σ is a map σ˜: P P satisfying → (1) σπ =πσ˜; (2) σ˜(p g)=σ˜(p) g for all p P,g U(n); · · ∈ ∈ whereg representstheentry-wisecomplexconjugateofg U(n). Weremarkthat, ∈ due to property 2 of a lift, the fixed point set Pσ˜ has the structure of a principal O(n)-bundle over the real points Xσ. Let σ˜ be a lift then we say that (P,σ˜) (X,σ) is a Real principal U(n)-bundle → (or Real bundle) if σ˜ further satisfies 3. σ˜2(p)=p for all p P; ∈ or if n is even we say that (P,σ˜) (X,σ) is a Quaternionic principal U(n)-bundle → (or Quaternionic bundle) if σ˜ satisfies 3′. σ˜2(p)=p ( I ) for all p P. n · − ∈ where I represents the n n identity matrix. Such bundles were classified in n × [BHH10]. HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER REAL SURFACES 3 Proposition 1.2 (Biswas,Huisman,Hurtubise). Let (X,σ) be a type (g,r,a) Real surface and denote its fixed components by X for 1 i r. Then Real principal i ≤ ≤ U(n)-bundles (P,σ˜) (X,σ) are classified by the first Stiefel-Whitney classes of → the restriction to bundles P X over the fixed components i i → ω (P ) H1(X ,Z/2)=Z/2 1 i ∈ i ∼ and the first Chern classes of the bundle over X c (P) H2(X,Z)=Z 1 ∈ ∼ subject to the relation c (P) w (P ) mod (2). 1 1 i ≡ Furthermore, given any such chaXracteristic classes then there is a Real principal U(n)-bundle that realises them. (cid:3) We write (c,w ,w ,...,w ):=(c (P),w (P ),w (P ),...,w (P )) 1 2 r 1 1 1 1 2 1 r and we will refer to the tuple (c,w ,w ,...,w ) Z Z as the class of the 1 2 r ∈ × r 2 Real principal U(n)-bundle (P,σ˜). Q Proposition 1.3 (Biswas, Huisman, Hurtubise). Let (X,σ) be a Real surface of type(g,r,a)andlet n beeven. Then Quaternionic principal U(n)-bundles (P,σ˜) → (X,σ) are classified by their first Chern class which must be even. Furthermore, given any such Chern class then there is a Quaternionic principal U(n)-bundle that realises it. (cid:3) We recall that we only cater for Quaternionic bundles of even rank. However,a similar result for the case when n is odd is also handled in [BHH10]. Writing c = c (P), we will therefore refer to c 2Z as the class of the Quater- 1 ∈ nionic principal U(n)-bundle (P,σ˜). Let (P,σ˜) (X,σ) be a Real or Quaternionic principal U(n)-bundle. An auto- → morphism of (P,σ˜) is a U(n)-equivariant map φ: P P such that the following → diagrams commute P φ // P P φ // P and σ˜ σ˜ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) X idX //X P φ //P. Let Map(P,P) denote the setofself maps ofP endowedwith the compactopen topology. Definition1.4. The(unpointed)gaugegroupG(P,σ˜)isthesubspaceofMap(P,P) whose elements are automorphisms of (P,σ˜). Itwillbeconvenienttoprovidedecompositionsforcertainsubspacesofthegauge group. Definition 1.5. Choose a basepoint of (X,σ) such that σ( ) = if r > 0. X X X Then the (single)-pointed gauge group∗G∗(P,σ˜) consists of the e∗lement∗s of G(P,σ˜) that restrict to the identity above . X ∗ 4 MICHAELWEST Another pointed gauge group of interest was considered in [BHH10]. Let (X,σ) be a Real surface of type (g,r,a), then for each 1 i r choose a designated ≤ ≤ point contained in the fixed component X . Further, if a = 1, choose another i i ∗ designatedpoint thatisnotfixedbytheinvolution. Byconvention,wechoose r+1 ∗ to be as in Definition 1.5. 1 X ∗ ∗ Definition 1.6. The (r+a)-pointed gauge group G∗(r+a)(P,σ˜) consists of the el- ements of G(P,σ˜) that restrict to the identity above these (r+a) designated points of (X,σ). 1.2. MainResultsforRealBundles. Inthissection,weaimtopresentthemain results pertaining to homotopy decompositions of gauge groups of Real principal U(n)-bundles. To ease notation we will sometimes use the following G((g,r,a);(c,w ,w ,...,w )) to represent the unpointed gauge group of a 1 2 r • Real bundle of class (c,w ,w ,...,w ) over a Real surface of type (g,r,a); 1 2 r G∗((g,r,a);(c,w ,w ,...,w ))to representthe single-pointedgaugegroup 1 2 r • of the Real bundle as above; G∗(r+a)((g,r,a);(c,w ,w ,...,w )) to represent the (r+a)-pointed gauge 1 2 r • group of the Real bundle as above. We first presentthe results relatingto whengauge groupsofdifferent Realbun- dles havethe samehomotopytype. For (r+a)-pointedgaugegroupsthis is always the case. Proposition 1.7. Let (P,σ˜) and (P′,σ′) be Real principal U(n)-bundles over a Real surface (X,σ) of arbitrary type (g,r,a), then there is a homotopy equivalence BG∗(r+a)(P,σ˜) BG∗(r+a)(P′,σ′). ≃ However, this is not necessarily the case for the single-pointed and unpointed gauge groups, although we do have the following results. Proposition 1.8. For any c,c′,w ,w′ there is a homotopy equivalence 1 1 BG∗((g,r,a);(c,w ,w ,...,w )) BG∗((g,r,a);(c′,w′,w ,...,w )). 1 2 r ≃ 1 2 r Proposition 1.9. Let the following be classifying spaces of rank n gauge groups. Then there are isomorphisms of gauge groups G((g,r,a);(c,w ,w ,...,w ))=G((g,r,a);(c+2n,w ,w ,...,w )). 1 2 r ∼ 1 2 r Proposition 1.10. Let n be odd then there are isomorphisms of rank n gauge groups (1) G((g,r,a);(c,w ,w ,...,w ))=G((g,r,a);(c, r w ,0,...,0)); (2) G∗((g,r,a);(c,w1 ,w2 ,...,wr ))∼=G∗((g,r,a);(c, i=1r iw ,0,...,0)). 1 2 r ∼ P i=1 i It would be better to provide stronger statements ofPPropositions 1.7 and 1.8, such as in the form of the isomorphisms of Propositions 1.9 and 1.10. Indeed, the proofs of the latter invoke a conceptually simple argument and it may be the casethatPropositions1.7and1.8canbe givenstrongerstatements usinga similar approach. We state homotopy decompositions for (r+a)-pointed gauge group in the form of Theorem 1.11. HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER REAL SURFACES 5 Theorem 1.11. Let (P,σ˜) be of arbitrary class then there are integral homotopy decompositions Type Decompositions for G∗(r+a)(P,σ˜) (g,0,1) for g even G∗((0,0,1);0) ΩU(n) × g (g,0,1) for g odd G∗((1,0,1);0) QΩU(n) ×g−1 Q (g,r,0) Ω2(U(n)/O(n)) ΩU(n) ΩO(n) ×(g r+1)+(r 1) ×r−1 − Q − Q (g,r,1) G∗((1,1,1);(0,0)) ΩU(n) ΩO(n) g−r even ×(g r)+(r 1)+1 ×r−1 − Q − Q (g,r,1) G∗((1,1,1);(0,0)) ΩU(n) ΩO(n) g−r odd ×(g r 1)+(r 1)+2 ×r−1 − − Q − Q In the single-pointedcasewe haveto be a little morecarefulwith regardsto the classof the underlying Realbundle. For the cases where G∗(r+a)(P,σ˜)=G∗(P,σ˜), 6 that is when r+a>1, we have the results in Theorem 1.12. Theorem 1.12. Let n be odd or let (P,σ˜) be of class (c,w ,0,...,0). Let r+a>1 1 then there are integral homotopy decompositions Type Decompositions for G∗(P,σ˜) (g,r,0) Ω2(U(n)/O(n))× Q ΩU(n)× QΩO(n)× QΩ(U(n)/O(n)) g−r+1 r−1 r−1 (g,r,1) G∗((1,1,1);(0,0))× QΩU(n)× QΩO(n)× QΩ(U(n)/O(n)) g−r even g−r r−1 r−1 (g,r,1) G∗((1,1,1);(0,0))× Q ΩU(n)× QΩO(n)× QΩ(U(n)/O(n)) g−r odd (g−r−1)+1 r−1 r−1 The remaining cases seem to integrally indecomposable, however we will obtain the following localised homotopy decompositions for odd rank gauge groups. Theorem 1.13. Let p=2 be prime and let n be odd, then there are the following 6 p-local homotopy equivalences (1) G∗((0,0,1);c) Ω2(U(n)/O(n)) Ω(U(n)/O(n)); p (2) G∗((1,0,1);c)≃ Ω2(U(n)/O(n))×Ω(U(n)/O(n)) ΩU(n); p (3) G∗((1,1,1);(c,≃w )) Ω2(U(n)/O×(n)) Ω(U(n)/O×(n)) ΩO(n). 1 p ≃ × × This resultreliesuponaselfmapofU(n)/O(n)asstudiedin[Har61],whichisa p-localhomotopyequivalenceif andonlyif n is odd. Hence, it seemsto be difficult to provide such satisfactory decompositions in the even rank case. We move on to some integral homotopy decompositions for unpointed gauge groups. The readeris invited to compare the tables of Theorem 1.14and Theorem 1.12. Theorem 1.14. Let (P,σ˜) be of class (c,w ,w ,...,w ) then there are integral 1 2 r homotopy decompositions 6 MICHAELWEST Type Decompositions for G(P,σ˜) (g,r,0) G((r 1,r,0);(c,w ,...,w )) ΩU(n) 1 r − ×g−r+1 Q (1) (g,r,1) G((r,r,1);(c,w ,...,w )) ΩU(n) 1 r g r even ×g−r − Q (g,r,1) G((r+1,r,1);(c,w ,...,w )) ΩU(n) 1 r g r odd ×g−r−1 − Q (2) G((2,1,1);(c,w )) G((1,1,1);(c,w )) ΩU(n). 1 1 ≃ × Further, for r 1 and when (P,σ˜) is of class (c,w ,0,...,0) or n is odd, there are 1 ≥ integral homotopy decompositions Type Decompositions for G(P,σ˜) (r 1,r,0) G((0,1,0);(c,Σw )) ΩO(n) Ω(U(n)/O(n)) i − ×r−1 ×r−1 (3) Q Q (r,r,1) G((1,1,1);(c,Σw )) ΩO(n) Ω(U(n)/O(n)) i ×r−1 ×r−1 Q Q (r+1,r,1) G((2,1,1);(c,Σw )) ΩO(n) Ω(U(n)/O(n)) i ×r−1 ×r−1 Q Q TheremainingunfamiliarspacesinTheorem1.14seemtobeintegrallyindecom- posable, however localising at particular primes permits further decompositions. Theorem 1.15. Let n be a positive integer and let p be a prime with p∤n. (1) Let the following be gauge groups of rank n then there are p-local homotopy equivalences (a) G((g,1,a);(c,0)) O(n) G∗((g,1,a);(c,0)); p ≃ × further if p=2 and n is odd, then there are p-local homotopy equivalences (b) G((0,06,1);c) SO(n) Ω2(U(n)/SO(n)); p (c) G((1,0,1);c)≃ SO(n)×Ω2(U(n)/SO(n)) ΩU(n). p ≃ × × (2) Let the following be gauge groups of rank p then there are p-local homotopy equivalences (a) G((g,1,a);(c,0)) O(p) G∗((g,1,a);(c,0)); p ≃ × further if p=2, then there are p-local homotopy equivalences (b) G((0,06,1);c) SO(p) Ω2(U(p)/SO(p)); p (c) G((1,0,1);c)≃ SO(p)×Ω2(U(p)/SO(p)) ΩU(p). p ≃ × × 1.3. Main Results for Quaternionic Bundles. To distinguish the notation of QuaternionicgaugegroupsfromtheRealcasewewilluseasubscriptQ,forexample G (P,σ˜). Further, to ease notation we will sometimes use the following Q G ((g,r,a);c) to represent the unpointed gauge group of a Quaternionic Q • bundle of class c over a Real surface of type (g,r,a); G ∗((g,r,a);c) to represent the single-pointed gauge group of the Quater- Q • nionic bundle as above; G ∗(r+a)((g,r,a);c) to represent the (r + a)-pointed gauge group of the Q • Quaternionic bundle as above. HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER REAL SURFACES 7 WepresentresultsinthesameorderaswedidintheRealcase. IntheQuaternionic case, the homotopy types of the pointed and (r + a)-pointed gauge groups are independent of the class of the bundle. Proposition 1.16. Let (X,σ) be a Real surface of fixed type (g,r,a). Let (P,σ˜) and (P′,σ′) be Quaternionic principal U(2n)-bundles over (X,σ), then there are homotopy equivalences (1) BG ∗(P,σ˜) BG ∗(P′,σ′); Q Q (2) BG ∗(r+a)(P≃,σ˜) BG ∗(r+a)(P′,σ′). Q Q ≃ For the unpointed case, we have an analogue of Proposition 1.9. Proposition 1.17. Let (X,σ) be a Real surface of fixed type (g,r,a) and let the following be gauge groups of Quaternionic bundles of rank 2n. Then for any even integer c, there is an isomorphism of topological groups G ((g,r,a);c))=G ((g,r,a);c+4n). Q ∼ Q We now present homotopy decompositions for pointed gauge groups in the Quaternionic case. The reader is invited to compare the following results to their Real analogues. Theorem 1.18. Let (P,σ˜) be a Quaternionic principal U(2n)-bundle of class c then there are integral homotopy decompositions Type Decompositions for G ∗(r+a)(P,σ˜) Q (g,0,1) for g even G ∗((0,0,1);0) ΩU(2n) Q × g (g,0,1) for g odd G ∗((1,0,1);0) QΩU(2n) Q ×g−1 Q (g,r,0) Ω2(U(2n)/Sp(n)) ΩU(2n) ΩSp(n) × g ×r−1 (g,r,1) for g r even G ∗((1,1,1);0) QΩU(2n) QΩSp(n) Q − × g ×r−1 (g,r,1) for g r odd G ∗((1,1,1);0) QΩU(2n) QΩSp(n) Q − × g ×r−1 Q Q Forthe caseswhereG ∗(r+a)(P,σ˜)=G ∗(P,σ˜), thatiswhenr+a>1,we have Q Q 6 the results in Theorem 1.19. Theorem 1.19. For (P,σ˜)of arbitrary class c, there areintegral homotopy decom- positions Type Decompositions for GQ∗(P,σ˜) (g,r,0) Ω2(U(2n)/Sp(n))× Q ΩU(2n)× QΩSp(n)× QΩ(U(2n)/Sp(n)) g−r+1 r−1 r−1 (g,r,1) GQ∗((1,1,1);0)× QΩU(2n)× QΩSp(n)× QΩ(U(2n)/Sp(n)) g−r even g−r r−1 r−1 (g,r,1) GQ∗((1,1,1);0)× QΩU(2n)× QΩSp(n)× QΩ(U(2n)/Sp(n)) g−r odd g−r r−1 r−1 8 MICHAELWEST Again, the remaining cases seem to integrally indecomposable, however we will obtain the following localised decompositions. Theorem 1.20. Let p=2 be prime, then there are p-local homotopy equivalences 6 (1) G ∗((0,0,1);0) Ω2(U(2n)/Sp(n)) Ω(U(2n)/Sp(n)); Q p (2) G ∗((1,0,1);0)≃ Ω2(U(2n)/Sp(n))×Ω(U(2n)/Sp(n)) ΩU(2n); Q p (3) G ∗((1,1,1);0)≃ Ω2(U(2n)/Sp(n))×Ω(U(2n)/Sp(n))×ΩSp(n). Q p ≃ × × We now present homotopy decompositions for the unpointed case. Theorem 1.21. For (P,σ˜)of arbitrary class c, there areintegral homotopy decom- positions Type Decompositions for G (P,σ˜) Q (g,0,1) G ((0,0,1);c) ΩU(n) Q g even × g Q (g,0,1) G ((1,0,1);c) ΩU(n) Q g odd ×g−1 Q (g,r,0) G ((0,1,0);c) ΩSp(n) Ω(U(2n)/Sp(n)) ΩU(n) Q ×r−1 ×r−1 ×g−r+1 Q Q Q (g,r,1) G ((1,1,1);c) ΩSp(n) Ω(U(2n)/Sp(n)) ΩU(n) Q ×r−1 ×r−1 ×g−r Q Q Q The remaining unfamiliar spaces in Theorem 1.21 seem to be integrally funda- mental, however localising at a particular prime permits further decompositions. Theorem 1.22. Let n be a positive integer and let p be a prime such that p ∤ 2n. Let the following be gauge groups of a Quaternionic bundle of rank 2n, then there are p-local homotopy equivalences (1) G ((g,1,a);c) Sp(n) BG ∗((g,1,a);c); Q p Q ≃ × (2) G ((0,0,1);c) Sp(n) Ω2 U(2n)/Sp(n) ; Q p ≃ × (3) G ((1,0,1);c) Sp(n) Ω2 U(2n)/Sp(n) ΩU(2n). Q ≃p × (cid:0) (cid:1)× (cid:0) (cid:1) 2. Proofs of Statements For the sake of clarity, we focus on the proofs of statements in the Real case, and then we elaborate on some of the details in the Quaternionic case in Section 2.5. We look to decompose the gauge groups by studying an equivariant mapping space as provided in [Bai14]. Throughout our analysis, we think of Real surfaces as Z -spaces. For Z -spaces 2 2 Y and Z, let MapZ (Y,Z) denote the space of Z2-maps from Y to Z. We note 2 that the fixed points of Y must be mapped to the fixed points of Z. If Y and Z are pointed, we denote a pointed version of this mapping space by Map*Z (Y,Z). 2 Further, recall the ‘basepoints’ of (X,σ) just before Definition 1.6. Let i ∗ A:= r+1 σ( ) ∐i=1 ∗i∐ ∗r+1 HOMOTOPY DECOMPOSITIONS OF GAUGE GROUPS OVER REAL SURFACES 9 andletMap∗Z(r+a)(X,Z)denote the subspaceofMapZ (X,Z)whoseelements send 2 2 A to 1. Let X denote the cofibre of A ֒ X and notice that there is a homeo- Z ∗ → morphism MapZ∗(2r+a)(X,Z)∼=Map*Z2(X,Z). A universal Real principal U(n)-bundle is given by (EU(n),ς˜) (BU(n),ς) → where ς is induced by complex conjugation and hence BU(n)ς = BO(n). Using this Z -structure, [Bai14] provides the following theorem. 2 Theorem 2.1 (Baird). There are homotopy equivalences (1) BG(P,σ˜) MapZ (X,BU(n);P); ≃ 2 (2) BG∗(P,σ˜) Map*Z (X,BU(n);P); ≃ 2 (3) BG∗(r+a)(P,σ˜)≃MapZ∗(2r+a)(X,BU(n);P)∼=Map*Z2(X,BU(n);P); where on the right hand side we pick the path component of MapZ (X,BU(n)) that induces (P,σ˜). 2 (cid:3) Thefollowinglemmacanbeshownbyadaptingtheproofinthenon-equivariant case. We will frequently require this lemma throughout the paper. Lemma 2.2. Let Y and Z be Z -spaces with basepoints fixed by the action, and 2 with Y locally compact Hausdorff. Then there are equivalences ∗ ∗ ((12)) ΩMMapa∗Zp2Z(X2(,XΩ,YY))∼=∼=MMaapp∗ZZ22((ΣΣXX,,YY)).; (cid:3) Throughoutthissection,thereareanumberofZ -spacesthatwilloftenappear; 2 here we provide a dictionary: (X,id) - any space X with the trivial involution; • (X X,sw) - the wedge X X equipped with the involution that swaps • ∨ ∨ the factors; (Sn, id) - the sphere Sn equipped with the antipodal involution; • − (Sn,he) - the sphere Sn equipped with the involution that reflects along • the equator. 2.1. Real Surfaces as Z -complexes. In order to provide homotopy decompo- 2 sitions for the gauge groups, it will prove useful to provide a Z CW-complex 2 structure for Real surfaces. The following is essentially a restatement of the struc- tures provided in [BHH10]. We let Σ denote a Riemann surface of genus p with p,q q open discs removed. Type (g,0,1). We first study the case where g is even. We can think of X as two copies of Σ glued along their boundary components; each a copy of S1. The g/2,1 involutionrestrictedtoS1 istheantipodalmapandextendstoswapthetwocopies of Σ . g/2,1 We give a CW-structure of X as follows, let X0 be 2 zero-cells; and σ( ). ∗ ∗ There are 2g+2 one-cells α ,...,α ,β ,...,β ,γ and 1 g/2 1 g/2 σ(α ),...,σ(α ),σ(β ),...,σ(β ),σ(γ). 1 g/2 1 g/2 1Ofcourse,itmaybenecessarytoassumethat∗Z isfixedbytheZ2-action. 10 MICHAELWEST Theboundariesofα ,β aregluedto andtheboundariesofσ(α ),σ(β )areglued i i i i ∗ to σ( ). One end of γ is glued to and the other to σ( ), whilst the same is done ∗ ∗ ∗ for σ(γ) with the opposite orientation. There are 2 two-cells glued on, one with attaching map α β α−1β−1 α β α−1β−1γσ(γ) 1 1 1 1 ··· g/2 g/2 g/2 g/2 andtheotherwiththesameattachingmapbutwithα ,β replacedwithσ(α ),σ(β ) i i i i and γσ(γ) replaced with σ(γ)γ. Asthenotationsuggests,theinvolutionswapscellsthatdifferbyσ. Inparticular, thisisaσ-equivariantCW-structureandhencedescendstoaCW-structureofX/σ. ′ Nowassumethatg isoddandletg =(g 1). We see thatX canbe thoughtof − as two copies of Σg′/2,2 glued along their boundaries; two copies of S1 in X. The involution swaps these copies of S1 but reverses orientations, and it extends to X to swap the two copies of Σg′/2,2. There are 2 zero-cells, and σ( ) and 2g one-cells ∗ ∗ α1,...,αg′/2,β1,...,βg′/2,γ,δ and σ(α1),...,σ(αg′/2),σ(β1),...,σ(βg′/2),σ(γ),σ(δ) whereα ,β ,σ(α ),σ(β ),γ,σ(γ)aregluedasbefore butthe boundaryofδ is glued i i i i to and σ(δ) to σ( ). Now there are 2 two-cells, one with boundary map ∗ ∗ α1β1α−11β1−1···αg′/2βg′/2α−g′1/2βg−′1/2δγσ(δ)γ−1 and the other gluedequivariantly. The cells δ and σ(δ) correspondto the copies of S1 above and here γ is a cell joining these copies of S1. Type (g,r,0). Lettheinvolutionfixrcirclesandletg′ =(g r+1)/2,thenX/σis − a Σg′,r and X can be thought of as two copies of Σg′,r glued along the r boundary components. Inthiscase,thebasepointispreservedunderσ,howeverX0 isgivenr zero-cells; one for each fixed component. The one cells are then α1,...,αg′,β1,...,βg′,γ2,...,γr,δ1,...,δr and σ(α1),...,σ(αg′),σ(β1),...,σ(βg′),σ(γ2),...,σ(γr) where α ,β are as before and γ joins the basepoint to the i-th fixed component i i i which is represented by δ . One of the 2 two-cells has attaching map i α1β1α−11β1−1···αg′βg′α−g′1βg−′1δ1γ2δ2γ2−1···γrδrγr−1 and we again define the other one equivariantly. Type (g,r,1) for r >0. Let the involutionfix r circles. We firstconsiderthe case where g r mod 2. Let g′ =(g r)/2, then X can be thought ofas two copies of ≡ − Σg′,r+1 glued along the boundary components. The involution fixes the first r of these components whilst restricting to the antipodal map on the extra copy of S1. Now, X0 is given (r+2) zero-cells ; one for eachfixed component and two for i ∗ the extra S1. The one cells are then α1,...,αg′,β1,...,βg′,γ2,...,γr+1,δ1,...,δr,δ and σ(α1),...,σ(αg′),σ(β1),...,σ(βg′),σ(γ2),...,σ(γr+1),σ(δ) where α ,β are as before and γ joins the basepoint to the i-th boundary circle. i i i Each fixed component is represented by δ and δ joins to and therefore i r+1 r+2 ∗ ∗