NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 125-140
INTEGRATION AND BORDISM ON SUPERMANIFOLDS
C. V ic t o r ia (Received July 1996)
Abstract. An integration scheme for Z2-forms on Berezin-Leites-Kostant su permanifolds together with a bordism theory is introduced. Its relationship with the berezinian integration is stated. Stokes formula is proved for this integration scheme, thus supplying a pairing between the cohomology of Z2- forms and bordism cycles. It is proved that the bordism groups of a given supermanifold depend only on its split (Batchelor) structure.
Introduction The de Rham complex of forms on manifolds was generalized in [28] to complexes of Z2-forms on real or complex differentiable Berezin-Leites-Kostant supermani folds, and a cohomology ring for them was obtained. In this paper we define an integration scheme for these Z2-forms (Section 3) and we prove that it verifies Stokes theorem (Theorem 4.2). Thus, the integration of cohomology classes of Z2-forms over bordism cycles of Berezin-Leites-Kostant supermanifolds - a no tion introduced in Section 5 - is well defined (Proposition 5.1). Whence, a pairing between cohomology and bordism is established. We also prove (Theorem 5.2) that the bordism groups, 0 (V)(X), of a supermanifold X are isomorphic to the bordism groups fi(.,.) (Gr(X)), where Gr is the functor which associates with any supermanifold its underlying linear supermanifold (also called its split (Batchelor) supermanifold). In order to better locate this paper in the existing literature, a few words should be said about the types of supermanifolds and the various relationships among them. Nowadays, the name supermanifold has basically two meanings: G-supermanifolds, and Berezin-Leites-Kostant supermanifolds (also called graded manifolds). G-supermanifolds are the closest to Rogers’ original theory of super manifolds [18]; they satisfy Rothstein’s axiomatics [21], and they contain several subcategories of supermanifolds (for instance, DeWitt, GH°°, H°° supermanifolds, etc.), which were sometimes considered as separate types of supermanifolds. The specific genealogy to which the general theory of G-supermanifolds gives rise to, together with some in-depth development of the various topics within this cate gory can be found in the monograph [3]. Berezin-Leites-Kostant supermanifolds, on the other hand, were introduced by Berezin and Leites in the East [4] [5] [13], and by Kostant in the West [12] (see also the book by Manin [15] for a thorough introduction to this theory). Even though there is a ono-to-one correspondence
1991 AMS Mathematics Subject Classification: 14F40, 57Q20, 58A10, 58A50, 58C35, 58C50. Key words and phrases: de Rham cohomology, Cobordism, differential forms, supermanifolds and graded manifolds, integration on manifolds; measures on manifolds, analysis on supermanifolds or garded manifolds. 126 C. VICTORIA between isomorphism classes of DeWitt supermanifolds and isomorphism classes of Berezin-Leites-Kostant supermanifolds, it is well-known that these categories are not equivalent because there are much more morphisms in the former than there are in the latter (see [3]). The framework for the integration and bordism schemes developed in this paper is that of Berezin-Leites-Kostant supermanifolds. A major problem in supermani fold theory is precisely that of defining a “suitable” integration theory (see problem (11) in [14]). As far as we know, there does not yet exist any general integration theory on G-supermanifolds. Nevertheless, there are some partial results. For in stance, the presentation of the berezinian sheaf made in [11] has been successfully used in [8] to construct a berezinian-like integration on DeWitt supermanifolds. Also, a path integration on Rogers supermanifolds has been developed in [19] [20]. On the other hand, some integration schemes have been proposed in the lit erature for Berezin-Leites-Kostant supermanifolds. The most popular one is the integration of compactly supported berezinian sections [4] [5] [13], along with its generalization for the non-compactly supported case [22]. Berezinian integration has been also used to define other volume sheaves: e.g., Berezinian integration is the origin of the pioneering work of [1] wherer | s-forms first appeared defined as objects whose restriction on compact (r, 5)-dimensional subsupermanifolds are sections of the berezinian sheaf (see also [29] [30] for a developement of these ideas); the complex of integral forms is defined as a tensor power of the berezinian sheaf with the dual of forms on supermanifolds. Thanks to this complex, a Stokes formula can be proved for the berezinian integration 6 []; the integral of the pseu dodifferential forms introduced in [7] coincides whit the berezinian integration on the (0, l)-tangent superbundles (see [7] and [15]). On the other hand, the integral of the superforms introduced in [9] is interpreted as the berezinian integral of the corresponding r | s-form by the Radon supertransformation (see [9] for details). The berezinian integration, however, does not satisfy the de Rham theorem, which constitutes one of its main drawbacks. Besides, its corresponding determinant - the berezinian - is not suitable for important algebraic operations such as Lagrange expansion. (See [26] for an alternative superdeterminant that does). There is also available an integration of 1-superforms along (1,1)-dimensional Berezin-Leites-Kostant supermanifolds introduced in [24] and used in [17] as an alternative to berezinian integration to perform calculus of variations. It does satisfy the de Rham theorem, but no generalization of it is known to higher dimen sions. This integration emerges from the theorem of existence and uniqueness for ordinary differential equations on Berezin-Leites-Kostant supermanifolds proved in [16]. This theorem generalizes and completes some results of [25] on integration of vector fields on supermanifolds (see problem (10) in [14] and also [15, Chp. 4, Section 4]). In [28] a Z2-graded de Rham cohomology functor - not determined by the usual de Rham cohomology of the underlying manifold - was introduced in the category of Berezin-Leites-Kostant super manifolds. It is an answer to problem (8) in the list of [14]. It is also to be said that a Z-graded, but not a Z2-graded, de Rham cohomology for G-supermanifolds can be found in [2] (see also [3]). In fact, for general G-supermanifolds this de Rham cohomology is not determined by the usual de Rham cohomology of the body either. However, this de Rham INTEGRATION AND BORDISM ON SUPERMANIFOLDS 127
cohomology of DeWitt supermanifolds is proved in [3] to be determined by the usual de Rham cohomology of its body. Thus, aside from the impossibility of a functorial comparison of the cohomologies of [28] and of [2] [3] (first because the categories of (isomorphism classes of) Berezin-Leites-Kostant supermanifolds and (isomorphism classes of) DeWitt supermanifolds are not equivalent, and moreover, because the cohomology in [28] takes values in a category of Z2-graded groups whereas that in [2] [3] does in Z-graded groups) the cohomology groups of a DeWitt supermanifold and of its corresponding Berezin-Leites-Kostant supermanifold are in general different. Here is now a brief description of the specific contents of this paper. In Section 1 we fix the notation, and recall from [28] the basic results on Z2-forms. In Section 2 we recall the more traditional description of integration of compactly supported berezinian sections (see [4] [5] [13] and also [15]) and the alternative description given in [11]. We also recall the integration of the non necessarily compactly case constructed in [22]. In Section 3 we define the integration of the Z2-graded forms on Berezin-Leites-Kostant supermanifolds (see Definition 3.3). This integration scheme does not use the berezinian integration at all, thus marking a major con trast with the integration theory for other types of forms on Berezin-Leites-Kostant supermanifolds introduced earlier in the literature (e.g., pseudodifferential forms, r | s-forms, superforms, etc.). Nevertheless, it emerges as a pleasant surprise the fact that the integration theory introduced here can in fact be related with the usual berezinian integration in the exact sense stated in Corollary 3.5. Stokes formula is proved in Section 4 (Theorem 4.2). This allows to define a pairing between the cohomology groups of Z2-graded forms constructed in [28] and the bordism groups introduced in Section 5. It must be mentioned that this pairing is different from that given for r | s-forms in [31] because of several reasons: Firstly, because the coho mology and the bordism groups used in [31] are defined only for the subcategory of Berezin-Leites-Kostant super manifolds with “proper” morphisms (in the sense of [29] [30]). By way of contrast, the cohomology introduced in [28] and the bordism introduced here, are well defined for the full category of Berezin-Leites-Kostant supermanifolds. Secondly, becasuse the restriction of the cohomology of Z2-forms to the subcategory of Berezin-Leites-Kostant supermanifolds with “proper” mor phisms is different from the cohomology of [29] [30], as it was proved in [28]. At any rate, one similarity is observed indeed: the bordism groups we introduce here are proved to depend only on the split class of the given supermanifold (see Theorem 5.2); a similar result holds true for the bordism groups of [31]. From now on all supermanifolds are assumed to be real differentiable (smooth) Berezin-Leites-Kostant supermanifolds. The contents of this work are part of the author’s Ph.D. dissertation [27]. The author deeply acknowledges the encouragement and support received by his advisor S. Xambo-Descamps. Special thanks are given to O.A. Sanchez-Valenzuela for all those helpful comments and criticisms. Last, but not least, the author would also like to thank the valuable suggestions made by the referee; they greatly contributed to the enlightening of the points made by this paper, by putting them into a wider perspective. 128 C. VICTORIA
1. Z2-Forms and Cohomology In the following paragraph we fix the notation to be used throughout this work. It essentially follows [15]and [29]. Let the ringed space X be a supermanifold of dimension (mo, m i), and let Ox denote its structural sheaf. We shall write Ox,[o) f°r its even part, and Ox,[i], for its odd part. The sheaf of nilpotents J\fx is Ox,\i] + {Ox,[i})2, and Tx denotes Afx/Al\. The underlying smooth manifold of X is denoted by Xred- It has the same underlying topological space as X and C^red =Ox/Afx- This yields a functor, (.)red, from su permanifolds to smooth manifolds. The identity of topological spaces, together with the projection Ox —► CxTed define the canonical closed embedding ix'- Xred —> X. Besides, lx is a locally free module of rank mi; it is locally generated by the classes of local odd coordinates on X. In this article we always assume that Xred is oriented. The underlying linear supermanifold Gr(X) (also called the split (or Batchelor) supermanifold) associated to X has the same underlying smooth mani fold as X and its structural sheaf Ogi-cx) is the sheaf of exterior algebras Aq^Tx- The identity between the underlying topological spaces and the natural inclusion ^Xred c ^Gr(X) define a canonical submersion PGr(X): Gr(X) —> Xred- It is well known that any smooth supermanifold is isomorphic (although non canonically) to its underlying linear supermanifold (see [15]).By a compact, resp. oriented, super manifold we shall always understand a supermanifold whose underlying manifold is compact, resp. oriented. If S' is a Z2-graded object (a superalgebra, a supermodule, a set, ...), then S[0], resp. 5[i], denotes the subobject of homogeneous elements of degree0 , resp. 1. The parity function As '■ S'jo]U We also use the parity functions defined on Z*, resp. on N, as where (mo, m i) £ N2 has been fixed. As it is usual in the literature, s denotes the parity, As(s), of s G S'. Unless stated otherwise, Einstein convention about summation of repeated in dices is assumed. For a pair (po,pi) £ N2, the (po,Pi)_tangent superbundle of a supermanifold X, T^Po,Pl)(X), was introduced in[28] by using the constructions of [23].Here we will only remember its construction in terms of local coordinates and we refer to [27]for a more intrinsic definition. The (po,Pi)~tangent superbundle of a supermanifold X is a fibered supermanifold on X, ^(Po,Pl)- T(P0,P1)(X) —> X, trivial over every open subsupermanifold U C X where a system of coordinates is defined. If X has dimen sion (mo, m i) then \poPl)(U) is the product U x (Mp°m°+pimi; Ojjpomo+pjmj X£, F G {1,... ,m0 + mi}, A G { ~ P i ,- 1 ,1 ,... ,Po}- The first group of cap ital letters A, B , ... are used to denote generic superindices in Z* and the second one F,G,... to denote generic subindices in {1,... , mo + mi}. So,A,B,... denote Az* (-^)? A^*B ),... ( and F , G,... denote A(mo mi^(F), A(m0jmi)(G ),..., and so on. The parity of coordinates X £ depends on their indices: X£ is even if F + A — 0 and it is odd if F + A = 1. Moreover, the parity j of small latin letters j G N means A(p0,pi)(i) where {po,Pi) is a fixed pair. The (po,Pi)~tangent superbunbles are functorial because any morphism of supermanifolds / : X —* ^ induces a morphism /* : T(P0,P1)(X) —> T(p°,pi)(y) of fibered supermanifolds which acts on the fibers by pulling back Y* to X £ • Let us quote from [28] the theorem which defines the complexes of 2Z-forms: Let be the subsheaf of (9x-supermodules of (7r(Po,p1))*^T(p0,pl)^x^ whose sections / can be locally written as J2Fi 'Fp0+pi (x)Xj,l • • • X ^X ^^ ... Xp^ where / Fl’ ,fpo+pi is a local section of Ox (/ is a homogeneous polynomial of de gree po+pi and of degree exactly 1 in each group of variables X 1,... , X Po, X -1, .. ., X~Pl). Theorem 1.1. Let X be a supermanifold of dimension (m0,m i) and (po,Pi) G N2. 1. There exists a morphism of sheaves of commutative groups ^ ( P0 ,P i): (7r(P0 ,P i))*^,T(P0 .Pi)(X) (7r(po + l,P i))*^>T(Po+i.Pi)(X) locally written as (- 1)po+piX£0+1 - (~1)AFX£ dx^ XA ^j- 2. There exists a morphism of sheaves of commutative groups ^(P 0 ,P i): (7r(P0,Pi))*^,T(P0.Pi)(X) ~ > (7r(po,Pi + l))* (^T(Po.Pi + i)(X) locally written as (- l)Po+PlX “ Pl_1 - (-1)AFX£ ^ ^ q x ^ ) ' 3. The sections f of which are solutions of the equations (no summation assumed) dX92f £dX § + v (- 7 ir+4(i+s)^SrirrdX*8X£ = o A ± B, A,B G {- p i,... , -1 ,1 ,... ,po}> F,G e {1,... ,m0 + m i}, define a subsheaf f2^ 0,Pl^of Ox~supermodules of on which D^po pi^+(1_a a) o D “p o ,p i) is z e r o >Va’fo G {°’ 1^ Proof. This is Theorem 4.2 of [28]. □ A Z2-form on X is by definition a section of ©(p0,pi)^x°’Pl^ They constitute a locally free sheaf of (9x-modules: Theorem 1.2. Let U C X be an open subsupermanifold where a system of coor dinates {a;F}is defined and let {X^}be the induced coordinates in the fibre of the 130 C. VICTORIA restriction of the tangent superbundles to U. Corresponding to these coordinates, define the homogeneous of parity A + F local differential operators d = x i- i- if6+**+vx tx ‘rgxs then the expression d Pl o...od * od£° o ... o d* (1) Xp-p1 Xf-1 XfP0 XF1 V ' which is obtained by applying the composition of differential operators d~Pl o ... o F-p i d r 1 o d?° o ... o d* to the constant function 1, where 1 ^ F\ ^ ^ FVn ^ J ' F _ i F PQ X ^ 1 F-1 ^ ... ^F-Pl ^ mo + mi and where the parity of any repeated index is 1, constitute a local basis for f2^ 0,Pl\ Proof. This is parts (1) and (2) of Theorem 4.11 in [28]. □ Using this description of Z2-forms, it was shown in [28] that the composition of differential operators induce a well defined product of Z2-forms. Thus, the sheaf of Z2-forms is in fact a sheaf of Z 2-graded (9x-algebras. The relation between the Z2-forms on a manifold Xred and its ordinary differen tial forms is the following Theorem 1.3. Let Xred be a manifold. 1. There exists a functorial isomorphism of OxTed-modules, ^(p^p*) : —> which locally assigns drrF_pi A ... A dxF-1 A d:rFpo A ... A dxFl to o ... o d ~ l o d p ° 0...0 d 1 (1). -p i 2. Both squares of the diagram r»(P o iP i) V'(P0,Pl) 0P0+P1 / P0’P l) rf(P o ,P i) a L*A-redy u u */Wed ' a u'Y* *^red d l D 1 o (p o + 1 ,p i) V-(P0+1’P l) . o P o + P i+ l > (p° ’pi +1) o (P0,P i+ l) iiiyXred ' db'y *^red * iuy *^red are commutative. is an isomorphism of complexes of algebras. Proof. Part (1) is Corollary 4.10 of [28]; for part (2) see the proof of Proposition 7.3 of [28]; part (3) follows from Remark 4.14 of [28]. □ Each one of the operators D° and D 1 in Theorem 1.1 defines corresponding (p0, p i)-cohomology groups of X as the quotient of the global closed Z2-forms by the exact global ones, that is INTEGRATION AND BORDISM ON SUPERMANIFOLDS 131 2. Berezinian Integration The purpose of this section is to motivate the definitions made in Section 3. For that we review the more traditional description of integration of compactly supported berezinian sections (see [4] [5] [13]and also [15])and the alternative description given in [1 1 ]. We also sketch the integration of the non necessarily compactly case proposed in [22]. All supermanifolds are here assumed to be ori ented. The berezinian sheaf of a supermanifold X is usually denoted as Berx and it is defined as (Ber f2^ddX)* (see [15]). It is locally free of rank 1, and a system of coordinates x\,... ,xmo+rni induces an isomorphism Berx Iu — Iu D*(drr) (notations as in [15]). A change of coordinatesx = x(y) with jacobian matrix transforms the generator D*(dx) into Ber D*(dy) where Ber is the berezinian of the jacobian matrix (we refer to[15] for the defintion of the berezinian of a matrix). If {:rF}, 1 ^F ^ rao+rai is a local system of coordinates with parities xf = A(mo mi)(F), any section 0 of Berx can be locally written as (xi , . . . , xmo XZmo + lT 1 ...(x mo+mi)amiD*(dx). ai,...,ami €{0,1} W ith these notations, the definition of the berezinian integral given in[15] is - Definition 2.1. Let 0 be a section of Berx with compact support in an open subset with coordinates a?i,... xmo+mi , . If the expression of 0 is ^ ^ (*El} i• ^TnoX^mo+l)• • 1 ••'(■*'mo+mi) mi D (dx), ai,...,Qmi €{0,1} then its berezinian integral is defined to be /•Ber /> / 0 = / • • • ,£ mo)dxiA ... A dxmo. J% JXred The intergral of a general berezinian section 0 with compact support is defined by additivity, by using a partition of the unity and the above local formula. The key result in order to guarantee that the berezinian integral is well defined is that the (9x-submodule BerxA/"™1 of Berx has a natural structure of Ox^-module and that with this structure, it is naturally isomorphic to ^x red ’ dim(X) = (mo> m i) (see [15]).Locally, this isomorphism assigns 0i,...,i(*i, • • •, Zm0)dziA ... A dxmo to 01,...,l(®l ,.. ., xmo )^mo+l • • 2• 'mo+mi D (da?). Moreover, the projection which sends ^ ^ 0ai...ami (®i5 • • • ?^mo)(a;rno-(-i) ... (xmo-).mi) miD (dx) Qi,...,ami €{0,1} to 01,...,1 (^1,%mo)x - - • , mo + l • • • X m o + m i ^ (dx) 132 C. VICTORIA defines a global projection Berx —► Berx .A/"™1. Thus, the berezinian integral of Definition 2.1 is the composition of these two morphisms followed by the usual integral on the underlying manifold: Berx -► BerxA/^1 -> ^red —> R and therefore it is well defined. Let us summarize the description of Berx in terms of differential operators given in [1 1 ]. One considers the quotient sheaf of the differential operators on X, dim (X) = (mo, m i), with values in modulo the differential operators P such that for any function / there exists a (mo — l)-form u on Xred which verifies (P (/))red = da;. It is shown that it is a rank 1 free Ox“module, locally generated by the class of d^i A ... Adxmc) dxi A ... A dxmo 3. Integration of Z2-forms In this section we will define three types of integration of Z2-forms (the berezinian integration, the integration as differential operators and the reduced integration), and we will show that in fact they are the same. All supermanifolds are here assumed to be oriented. Berezinian integration of Z2-forms. As it has been remembered in Section 2 the integral of a berezinian section on X with compact support is the usual integral on Xred of the mo-differential form corresponding to its projection onto Berx-A/'x'1- Let us show that Z2-forms can be mapped onto Berx A/"™1 and therefore that it is possible to define the berezinian integral of a compactly supported Z2-form as the berezinian integral of its image in BerxA/"^1. The well known fact that a supermanifold is (not canonically) isomorphic to its underlying linear supermanifold can be rephrased by saying that there always exists an adapted atlas which by definition is an atlas such that if x±,... , xmo, £ i,... , £mi INTEGRATION AND BORDISM ON SUPERMANIFOLDS 133 and 2/1,... , ymo, Vh ■ • • are two system of local coordinates (the latin letters are even coordinates and the greek one are odd) then the even coordinates yF's depend only on the even coordinates X q s and the ryF’s are linear combinations of the £G’s whose coefficients areC°° functions on x i,... ,Xm0■ To deal with the berezinain integration of Z2-forms we assume that an adapted atlas has been fixed, and we denote by p% the composition of the isomorphism X —> Gr(X) with the canonical submersion PGr(X)- Therefore the operators dyF depends on d£G but not on d^ /, concretely dA = y d* V f Z -/ x g l^G^mo ° As a consequence the local expressions d”^1 o ... o d~^ ^ o d ^ o ... o d*Fp (1) with 1 < Fi < ...F Po < F _i < ... < F_Pl ^ mo constitute a local basis for an C?x_submodule of The composition j/ P0,Pl) of the projection followed by the morphism co%0’Pl^ —> P x i^x ^f1) which locally assings g(x,£)®dxF_pi A .. .Ad:rFl to g(x, £)d“Pl °- • -°diFl (1) is an epimorphism. Moreover, is related with the berezinan sheaf as follows - Lemma 3.1. Let denote the dual of the local basis £i ... £mifor A/"™1 as Oxred~ module. There exists an epimorphism of O-x-modules on the left P x W Z J ®o Xm (A/£ ')* - Berx which locally assigns hgD* (d(x, £)) to h® dxi A ... A dxmo ® gd£. Proof. It is easy to verify that the transition functions of the sheaf on the left corresponds by the local isomorphisms to the transition function of the berezinian, D As a consequence, by tensorializingfi by 1 we have P 'x(»T J-» BerxATJ' c Berx. Definition 3.2. The berezinian integral of a (po>Pi)-formu with compact support and such that Po + Pi = mo is defined to be its image by the composition f ®er 0 0 jg > ~ )^ pin^^B erx^K P o + P i= m o po+ p± =m o that is The reduced integral of 7?-forms. In [28](Theorem 4.5) it is shown that Z2- forms are functorial with respect to any morphism of supermanifolds. In particular one can consider the pullback with respecto to the canonical closed embedding : Xred —*”^ and then the following definition makes sense Definition 3.3. The reduced integral of Z2-forms with compact support is the composition •* r»(PO iPl) *3C. o (P O iP l) r^P0+ Pl Xred 134 C. VICTORIA Notation. For short, we will write uj |xred instead of (ip(Po,Pl) ° ^xX^)* This reduced integral of Z2-forms is more canonical than the berezinian integral, because does not assume that any isomorphism X ~ Gr(X) has been fixed. Integration of 7?-forms as differential operators. In Section 2 we have reviewed the integrations of [11]and [22].The volume sections in both cases are essentially tensors products a® P in which a is a top degree differential form on the underlying manifold and P is a differential operator on X. The integral of such a volume section is the ordinary integral of the contraction of the Fermi integral of P (which is by defintion, see [22], the value ofP on the constant function 1, P (l)), with the differential form a, fx (a P (l))red. Using this language, we know from Section 1 that Z2-forms are the Fermi integral P (l) of certain differential operators P on the tangent superbundles. Thus by analogy with the integrals of [11]and [22] one can ask if the expression f x d (P (l))red can be understood in any sense as an integration of Z 2-forms. In the proof of Proposition 3.4 it is shown that this is the case, concreately that if P (l) is a Z2-form, then P (l)reci = P( 1) |xred- Proposition 3.4. When an isomorphism between the supermanifold and its un derlying linear supermanifold is fixed, the three preceeding integrals are the same. Proof. Let x i,... , xmo+mi be a system of x i,... , xmo even and xmo,... , xmo+rni odd coordinates of an adapted atlas on X and let cjF_pi ’"'’F_1,Fpo’ ”’Fl (x)d“Pl o * —pi ... o d~F o dP(^, o ... o dXFi (1) be the local expression of a (po,Pi)_formuj with Po + Pi =rn0. Then: rBer ( / 0 !>»•«> \ PO+ Pi=m 0 Ber Xmo + 1 ■ ■ ■ Xmo+rm ^ ^ (x)D*(dx) Fi :Fi= 0 y ] «x(u;F_pi ’"'’F_1’Fp°’'",Fl(:E))da:F_pi A ...dx AF_1 A d:rFpo A ...dxFl. A Fi :Fi= 0 On the other hand, the image by of the pullback of uj by zx is ^ i%(ujF~pi" ’F-1,Fpo'"Fl(x))dxF_pi A ...dxF_1 A A dxFpo A ...dxFl A Fi :Fi— 0 and therefore . uj |x,.ed coincides with the berezinian integral ofuj. The Z2-form uj = ^ F.a;F-pi --«F-1-Fpo>"-Fl(x)d-P*_pi o ... o d " ^ ° d ^ o ... o d J F i(l) is the Fermi’s integral of the differential operator P = Y^FiujF~P1'"''F~1'Fpo'"',Fl(x)d~Pl o ... O d~l_^ O dP°Fpo o ... o d iFi. Therefore, (P (l))red is in fact uj |Xred, and the ex pression (P (l))red proposed above is precisely Jx d uj |xred- d As a consequence the dependence of the berezinian integration of Z2-forms on a fixed non canonical isomorphism between the supermanifold and its underlying linear supermanifold is only apparent. INTEGRATION AND BORDISM ON SUPERMANIFOLDS 135 Rem ark. It is to be observed that in [11] and [22] the volume sections are differen tial operators P tensorialized with differential formsu j of the underlying differential manifold, u j 0 P. As a result of the Fermi integral of such a section one obtains a form uiP{ 1) whose reduced part is integrated as usual on the underlying manifold. In contrast, to integrate a Z2-form u; which is the Fermi integral of a certain differ ential operators (see Theorem 1.2), u j = P( 1), there is no need of tensorializing it with differential forms of the underlying differential manifold. This is because, as we have seen in Section 1, ordinary differential forms can be viewed as the Fermi integral of certain differential operators using the isomorphism and therefore they naturally appears as ‘components’ of P (l). It is to be observed that although the morphism (/x 0 ■^x1)0^ © po+p1=m0 ^ Po,Pl is not onto, we can consider it as an epimorphism from the integration point of view: Corollary 3.5. Let (3 be a berezinian section with compact support, then there exists a 1?-form ;u with compact support such that the integral of (3 is the integral o f UJ. Proof. If the integral of /? is zero then we can obviously takeu j = 0 and if it is not zero, thenj3 is necessarily a section of the subsheaf BerxA/’^1 of Berx, as it is proved in [15]. Therefore, by Lemma 3.1 there exists a Z 2-formu j such that and then fxu = f* er (n 0 Af^1) o ( © po+pi=mo !/(*>,pO) (w) = /®er (3. □ 4. Stokes Formula In this section we will briefly recall the versions of Stokes formula of [15] [22] and [31]and we will state Stokes formula for the Z2-forms introduced in [28]. All supermanifolds are assumed to be oriented. As far as we know there exists in the literature four generalizations to super manifolds of Stokes formula predating the one we will give in this section: Stokes formula for integral forms [6],the Stokes formula proved in [31]for the forms in troduced in [1] [29] [30], Stokes formula for Rothstein volume sections22] and[ the Stokes formula proved in [9] for superforms. Stokes formula for integral forms. The Spencer complex of the berezinian, (Berx0e>x S'(TXII), 5), is proposed in [6](see also [15])as a generalization of the de Rham’s complex. The differential operator 8 increases by 1 the degree of the integral forms of degree —i, which by definition are the sections of Berx 0 o x d' : ^Xred is defined by the formula d'co(f) = d(w(/)). The complex so obtained is not functo rial with respect to arbitrary morphisms of supermanifolds. Nevertheless, a notion of boundary of a supermnaifold 0 : dX —► X can be defined for which there exists the pullback of sections of and such that one has the following version of Stokes formula /•Ro />Ro / d'u: = / 0*o;. Jx Jy Stokes formula for superforms. Since it is not closely related with our topics we refer the interested reader to [9]. After this summary, let us now state Stokes formula for Z2-forms: Stokes formula for Z2-forms. First of all we need a suitable definition of boundary of a supermanifold: Definition 4.1. A boundary of a supermanifold X is any commutative diagram <9X * > X ix ( ^red i ry* (CAA/Jred * ^red where 0red is the identity onto d(Xred)- The following theorem states that Z2-forms verifies Stokes formula with respect to the integration introduced in Section 3: Theorem 4.2. fx Daco = a = 0,1. INTEGRATION AND BORDISM ON SUPERMANIFOLDS 137 Proof. We write D for Da, a = 0,1 and ip instead of the functorial isomorphisms ^(po,pi) 0f part; 2() in Theorem 1.3. / D u ; = / ipi%Duj= / il)Di*xuj= / dipiyU / = (f)*redipi*xuj = JX JXred ''Xred J Xied J d(Xred} = Jd(Xr^)[ ^red*XW = J [ d(XTPd) *l>i*dX 5. Bordism The concep of bordism groups can be easily generalized to supermanifolds. The Stokes formula for Z2-forms proved in Section 4 allows to define a pairing between the bordism groups of a supermanifold and the cohomology of its Z2-forms. More over we will see that the bordism groups of a supermanifold depends only on the underlying linear supermanifold (Theorem 5.2). The definition of the bordism group of degree (po,pi) of a supermanifold X we will use here is a straightforward generalization of Thom’s definition of these groups for manifolds as it is presented in [10]. We will denote a supermanifold over X, / : y —» X, such that yred is compact and oriented, by a pair (y, /). Associated with (y, /) one can consider the pair (~y,f) where — y denotes the supermanifold y whose underlying manifold (—^)red has the opposite orientation of yred■ (V,/) is said to be cobordant to zero on X if there exists a pair (Z,g), with £ red compact and oriented and a commutative diagram V) X Z — ^ X where j : y —> Z is a boundary of Z in the sense of Definition 4.1. Two pairs ( ^ , /;), i = l,2 are defined to be cobordant over X if the disjoint union (^i, /i) U (—y2, / 2) is cobordant to zero on X. The supermanifolds over X of dimension (po5Pi) modulo the relation of being cobordant constitute the bordism group of degree (po,Pi) of X, fi(po,pi)(X), where the sum is the operation induced by the disjoint union. A cycle on X is a pair (y, f ) with empty boundary, dy = 0 . The cycles on X of dimension (Po,Pi) form a subgroup fi(P0)Pl)(3C) of ft(p0jPl)(X). Proposition 5.1. 1. If (^ i,/ i) and (^25/2) are cobordant and uj is a closed Z2-form (either for D° or for D1), then [ / i » = [ / 2» . Jy1 Jy2 2. If (y, /) is a cycle, then Daf*(cu) = 0, a = 0,1 for any I?-form u on X. Proof. Because Z2-forms are functorial (see [28]), the proof of this proposition is a straightforward generalization of the proof of the analogous statement for manifolds (see [10]). □ 138 C. VICTORIA As a result of this proposition the integration of Z2-forms defines a pairing n (p„,p0 (X)® RH D”,Pl)(X ) ^ R [y ,/]® W fy /*(<*>)• As it was said in the introduction of this article, this pairing and the similar one in [31] are different for two reasons: the cohomology and bordism groups involved in [31] are defined in the category of supermanifold with proper morphisms (in the sense of [30]), which is a strict subcategory of the category of supermanifolds, and moreover even if our constructions are restricted to this subcategory, the cohomol ogy of Z2-forms is proved (Section 4 of [28]) to be different from the cohomology groups given in [1] [30] [29], The following theorem states that the bordism groups depend only on the un derlying linear supermanifold of the supermanifolds. Theorem 5.2. 1. If there exists a homotopy y x [0,1] —> Xbetween the morphisms f,g: y —» X, then (y,f) and (y,g) are cobordant. 2. (y,f)and(y,g) are cobordant ifand only if (Gr(V), Gr(/)) and (Gr(y), Gr(g)) are cobordant 3. The groups f2(P0)Pl)(X) and ^(Po,Pl) (Gr(X)) are isomorphic. Proof. The first statement follows from the fact that (y, /) and (V, g) are cobordant by using the homotopy y x [0,1] —» X between / andg. To see the direct implication of the second claim, it is enough to apply the functor Gr. The reciprocal follows from the fact that any morphism / of supermanifolds is homotopic to Gr(/) (see [29]). The third statement is a corollary of (1) and (2). □ As it was said in the introduction of this article, the bordism groups of superman ifolds constructed in [31] also depend only on the underlying linear supermanifold (see [31]). References 1. M.A. Baranov and A.S. Shvarts, Cohomology of supermanifolds, Functional Anal. Appl. 18 no. 3 (1984), 236-237. 2. C. Bartocci and U. Bruzzo,Cohomology of supermanifolds, J. Math. Phys. 28 (1987), 2363-2368. 3. C. Bartocci, U. Bruzzo and D. 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Local coordinates {xF},F € {1,... ,mo + m i} of a superman ifold of dimension (mo,mi) defined in the open set U are assumed to be labeled so that A(xf ) = A(mo,mi)(F). Such a system of coordinates induces coordinates of the fibres over U denoted by capital letter with subindices and superindices INTEGRATION AND BORDISM ON SUPERMANIFOLDS 129o-r ^ x ■ It is a locally free sheaf of 0 x“modules with local basis