Digital Object Identifier (DOI) 10.1007/s00220-007-0378-0Commun. Math. Phys. 277, 715–727 (2008) Communications in

MathematicalPhysics

Torsion Cycles as Non-Local Magnetic Sourcesin Non-Orientable Spaces

Marcos Alvarez

Centre for Mathematical Science, City University, Northampton Square, London EC1V 0HB,UK. E-mail: m.alvarez@city.ac.uk

Received: 8 January 2007 / Accepted: 19 April 2007Published online: 7 November 2007 – © Springer-Verlag 2007

Abstract: Non-orientable spaces are known to be able to support net magnetic fluxesthrough closed surfaces, even in the absence of magnetic sources. To an observer forwhom the non-orientability of space appears to be confined to a certain finite region ofspace, such a magnetic flux appears as a magnetic charge quasi-localised in that region.In this paper it is shown that this effect is a physical manifestation of the existence oftorsion cycles of codimension one in the homology of space.

1. Introduction

The idea that electromagnetic charge may be a manifestation of the topology of space hasa very long history [Whe62, MW57, Lub63, Sor77]. The basic configuration consideredby Wheeler and Misner was a space that contained a handle-like region, often calleda “wormhole”. This space is traversed by sourceless electric lines of force that enterthe wormhole radially through one mouth and exit through the other. For an observersituated sufficiently far from the wormhole, space appears approximately flat, but notsourceless: the two mouths look like two pointlike electric charges.

A “dual” configuration can be constructed in which the lines of force are orientedcircles that surround the wormhole mouths and extend to its interior in a continuous way,as sketched in Fig. 1. From a distance, the two mouths appear to carry dual charges ofopposite signs. As a matter of terminology, we will refer to the dual lines of force, andthe charges they correspond to, as magnetic, although the motivation for doing that willonly become clear later on.

Serious objections can be raised against Misner and Wheeler’s wormhole as a realis-tic model of electromagnetic charge, on the grounds that wormholes are not satisfactorysolutions in General Relativity. Under fairly general assumptions, the null energy condi-tion has to be violated at some points near the wormhole throat [HV97]. This indicatesthat wormhole solutions require the coupling of gravity to some type of exotic matter.Even if the right type of matter and coupling is found, known wormhole solutions are

716 M. Alvarez

Fig. 1. An orientable handle

very unstable. Charged wormholes also exist in General Relativity in the presence ofscalar fields non-minimally coupled to gravity. The scalar field provides a good exampleof the type of exotic matter required, but these charged wormholes are still unstableunder spherically symmetric perturbations [BG03].

Another unsatisfactory feature of the Misner-Wheeler construction is that it is unableto accommodate individual isolated charges. Besides, the throat that joins the two appar-ent charges establishes a correlation between them that is difficult to reconcile withquantum mechanics [Sor77]. For these reasons, the Misner-Wheeler wormhole cannotbe taken seriously as a realistic model of charged particles.

Some time later, Sorkin [Sor77] constructed a non-orientable version of the Wheeler-Misner wormhole, and showed that the effect of the non-orientability was that, to a distantobserver, the non-orientable handle appears to be a source of net flux.

Figure 2 gives a two-dimensional illustration of this idea. A non-orientable surfaceis crossed by magnetic lines of force, and the non-orientability causes the lines of forceto emerge with the same orientation from both ends. A distant observer would assignthe same magnetic charge to both of them. In contrast, electric flux lines would havebehaved very much as in Fig. 1, and each end would have seemed to still carry oppositeelectric charges. This example suggests that the non-orientability of space affects onlyone kind of charge, namely the one we called dual or magnetic.

The construction proposed in [Sor77] covers a large class of non-orientable mani-folds, not limited to handles, and provides a model of individual, isolated charges thatis based purely upon the topology of space. These ideas were further elaborated andextended in [DH99], who generalised Sorkin’s proposal and provided new illustrationsof it.

The ideas that will be discussed in the main body of this paper are similar to thosein [Sor77], although the point of view will be rather different. None of the objectionspointed out earlier will be answered, and it will not be claimed that these models canprovide an alternative model of electromagnetic charge. The goal will be to show thata different connection between magnetic charge and non-orientability can be made, inwhich the topological properties of non-orientable spaces appear more explicitly.

2. First Notions

We shall be working in an (m + 1)-dimensional spacetime of the form R × K, wherethe real line R is the time direction. By construction, this spacetime is time-orientable,

Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces 717

Fig. 2. A non-orientable handle

so it will be orientable if and only if the space manifold K is. In this spacetime thereis defined a closed (m − 1)-form F , the electromagnetic field strength. The presenceof magnetic or electric sources is expressed by the dual currents µm and µe, which arerelated to the field strength F and its Hodge dual ∗F by the equations,

d F = µm,

d∗F = µe.(1)

The forms µm and µe are taken to be purely spatial, i.e., are dual to timelike forms.This is a physical assumption based on the fact that the world-volumes of localisedcharged sources are always timelike. The dual current, being transversal to the world-volume, must then be always spacelike.

Assuming that the Stokes theorem for differential forms can be applied, (1) relatethe magnetic and electric charges to the fluxes of F and ∗F respectively. Suppose thatN is an m-dimensional region of K whose frontier is a smoothly embedded compact(m − 1)-manifold S. Then the magnetic charge contained in N ,

Qm(N ) =∫

Nµm, (2)

equals the flux of F through S,

Qm(N ) =∫

Nµm =

∫

Nd F =

∫

SF.

Similarly, if ν is a 3-dimensional region of K bounded by σ , the electric charge containedin ν, defined as

Qe(ν) =∫

ν

µe, (3)

equals the flux of ∗F through σ ,

Qe(ν) =∫

ν

µe =∫

ν

d∗F =∫

σ

∗F.

718 M. Alvarez

Although charges can be defined directly in terms of the currents, as in (2) and (3), wewill only be concerned with those charges that can be expressed as fluxes. The generalquestion of whether charges are always fluxes or not depends on certain relationshipsbetween the homologies of the world-volumes of the sources and of the spacetime mani-fold in which they propagate. For a recent analysis of this question in compact orientablemanifolds with boundary, see [AO06].

3. Global Considerations

Equations (1), which hold locally at every point in space, leave open the possibility thatF or ∗F , or the dual currents µm and µe, may not be globally defined [Kis78]. Weshall assume that the (m − 1)-form F is global, i.e., single-valued. In our model, thedual magnetic current µm is an m-form on the m-manifold K. Consequently, it must beproportional to ΩK, the volume form of K,

µm = C(w)ΩK, (4)

in which C(w) is a distribution localised on w, the world-line of the magnetic charge.Equation (4) only makes sense globally if K is orientable. If it is not, there exists a loopλ along which an ΩK chosen at one point gets transported back into−ΩK, which wouldbe interpreted as the magnetic charge having changed sign after traversing λ.

Because we are assuming that F is single-valued, the same must be true of µm . Inwhat follows, K will be non-orientable, and so, as discussed, µm can only be definedglobally up to a sign. This leaves µm = 0 as the only possibility compatible with ourassumptions. That makes F a closed form, i.e., the first equation in (1) reduces to d F = 0.No nonzero single-valued magnetic charge, as defined in (2), can exist in K.

Another implication of K being non-orientable is that, because there is no global vol-ume element with which to define the Hodge star, the dual field strength ∗F cannot existglobally. It will reverse sign whenever ΩK does, as was the case when it was transportedaround the loop λ. Therefore µe will also have to change sign upon traversing λ.

In conclusion, within the framework of the equations of motion (1), spacetimes ofthe form R × K, where the spatial manifold K is non-orientable cannot support non-zero single-valued electric or magnetic currents, and hence neither electric nor magneticcharges in the sense of (2) and (3).

It will be shown below that, despite the exclusion of magnetic sources implied bythe condition d F = 0, it is still possible to define a different kind of magnetic charge,understood as a flux of F through a cycle in K, which owes its existence precisely to thenon-orientability of the space manifold K.

4. Fluxes, Orientations, and Charges

Given the closed differential (m − 1)-form F and a smoothly embedded orientable(m − 1)-cycle S in K, the magnetic flux Φ(S) is defined as

Φ(S) =∫

SF. (5)

The assumption of orientability of S is needed in order to integrate the differential formF on S [AMR88, BT82]. Let us now add the extra assumption that S separates the

Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces 719

space manifold K into two parts that we call M and N , with M non-compact andorientable, and N compact, possibly non-orientable, and has S as its only frontier. AnN non-orientable will be interpreted below as a quasi-localised region of space whosetopology is capable of displaying magnetic charge, as first suggested by Sorkin [Sor77].Configurations of this type can be constructed by taking the connected sum of twom-manifolds, one non-compact and orientable and the other closed and non-orientable.The connected sum consists in removing the interior of an embedded m-ball from eachmanifold, and pasting the remainders together by means of an homeomorphism on theboundary spheres of these balls. The identification of the two boundary spheres givesthe cycle S.

If there is net magnetic flux Φ(S), we will say that N carries F-charge.We now proceed to relate Φ(S) to the behaviour of F in the interior of N by means of

the Stokes theorem. In fact, there are two versions of the Stokes theorem, depending onwhether we are working with differential forms, or with densities [AMR88, BT82]. Inte-grating differential forms requires an orientation with which to turn the volume elementinto a measure. Hence, the Stokes theorem for differential forms is only applicable toorientable manifolds, and in particular we need N to be orientable. If so, then∫

Nd F =

∫

SF. (6)

Remembering that F is closed, it follows that an orientable region N never carriesF-charge,

N orientable =⇒ Φ(S) = 0. (7)

The question that will occupy us in the rest of this paper is whether the F-charge can benon-zero if N is non-orientable, and what its values can be.

The Stokes theorem for differential forms (6) is inapplicable when N is non-orient-able, so no conclusions relevant to our question can be drawn from it. Nevertheless, wewill find a way to make sense of (6) for N non-orientable by means of a cell decomposi-tion, as will be explained below. This procedure will relate the values of Φ(S) to certaintopological quantities intrinsic to N , independent of the choice of cell decomposition.

5. Two Notions of Boundary

In point-set topology, an m-manifold with boundary is locally mapped to R+ × Rm−1,and the boundary is the submanifold that gets mapped into 0 × Rm−1. That conformsto the ordinary idea of boundary as a frontier where the manifold ends, and in fact wehave been referring to this notion as “frontier” in previous paragraphs.

From the point of view of homology, an m-manifold can be regarded as a sum ofm-cells, each cell being homeomorphic to an m-ball. The m-cells can be visualisedas m-dimensional polyhedra that fill out the entire manifold without overlapping. Theboundary of an m-cell, defined as in point-set topology, i.e., as its frontier, is now regardedas the sum of its faces. These faces now are the (m − 1)-cells in our cell decompositionof the manifold, and so on.

The collection of all k-cells is known as the k-skeleton of the manifold, where k runsfrom 0 to m. We will adopt the notation V k

a for the elements of the k-skeleton, where ais an appropriate index that labels its elements. Because the cells V k

a are homeomorphicto k-balls, the boundaries ∂V k

a are homeomorphic to (k − 1)-spheres.

720 M. Alvarez

The precise way in which the boundaries of the (r + 1)-cells fit into the r -skeleton issummarised in the incidence matrix Mr+1,r

ab , defined by

∂V r+1a =

∑b

Mr+1,rab V r

b . (8)

The entries of these matrices can only be 0 or ±1. As a matter of terminology, we willsay that V r+1

a incides upon V rb whenever Mr+1,r

ab = 0.The operation of taking the boundary of a cell is required to be nilpotent, that is,

it must always be the case that ∂2 = 0. At the level of cells, this requirement followsfrom the earlier statement that ∂V m

a is homeomorphic to the sphere Sm−1, and henceboundaryless. In order to reflect this nilpotency, the incidence matrices are required tosatisfy the relationship

∑b

Mr+1,rab Mr,r−1

bc = 0. (9)

In the definition of the m-dimensional cells V ma , there is implied a choice of orientation

for each one of them. Recall that the m-cells are homeomorphic to m-balls, and for thesethe notion of orientation is straightforward. Then, just as for an m-ball, there are twochoices possible for the orientation of each cell. Although there is no preferred way tochoose all the orientations, some choices are better than others in a sense to be explainedbelow. For the time being, we will assume that every m-cell has been given a definiteorientation. This induces an orientation on its boundary in the usual way. Once this isdone, the integration of differential forms is defined both in the interior of the cells andon their boundaries.

Now, we identify the manifold with the sum of all the m-cells, N =∑a V m

a . Whenwe combine this identification between the manifold and its cell decomposition with thedefinition (8) of the incidence matrix, the boundary of N can be given as

∂N =∑

a

∂V ma =

∑a

∑b

Mm,m−1ab V m−1

b . (10)

Every (m − 1)-cell in the cell decomposition of a manifold is the face of one or twom-cells. In contrast, cells of dimension m−2 and lower can be shared by any number ofhigher-dimensional cells, depending on the details of the cell decomposition. It followsthat the colums of the incidence matrix Mm,m−1

ab can be of three types:

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

V m−1a, type I

0

.

.

.

010

.

.

.

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

or

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

V m−1a, type II

0...

010...

0−1

0...

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

or

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

V m−1a, type III

0...

010...

010...

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces 721

up to overall signs that can be arranged to be as shown. The first type corresponds to cellsV m−1

a, type I that lie entirely on the frontier of N . The second and third types correspond to

cells V m−1a, type II or V m−1

a, type III that are shared by two different m-dimensional cells. Thedifference between these two types becomes apparent when we use (10) to calculate∂N . All the type II cells cancel out, and we are left with

∂N =∑

a

V m−1a, type I + 2

∑a

V m−1a, type III. (11)

This formula illustrates the difference between the notions of boundary used in homol-ogy theory and point-set topology. In our notation, the latter comprises all the type Icells, and nothing else; as a subspace of N , it coincides with the frontier S. The formerincludes also an (m − 1) cycle made of all the type III cells, multiplied by two. It willbe explained below that the appearence of this cycle is due to the existence of a torsionsubgroup in the homology group Hm−1(N ,S;Z) that is cyclic of order two wheneverN is non-orientable.

Lacking a standard terminology, we will use the name “free boundary” for the point-set boundary, and the name “torsion boundary” for the contribution of the torsion (m−1)-cycles. The “homology boundary” is the sum of both. The symbol ∂N will be reservedfor the homology boundary of N , and so we write

∂N =∑

a

V m−1a, type I + 2

∑a

V m−1a, type III = (∂N )free + (∂N )torsion. (12)

As a chain, the free boundary is the sum of all type I cells, and so it manifestly dependsupon the choice of cell decomposition. As a subspace of N , however, it does not, because,as has already been pointed out, it coincides with the frontier S. In contrast, the torsionboundary does depend on that choice, even as a subspace of N . It will now be shownthat, if N is orientable, then there always exists a cell decomposition for which there isno torsion boundary.

6. Orientability and Cell Decompositions

When we calculate ∂N using an arbitrary cell decomposition, the result is generally ofthe form

∂N = (∂N )free + 2γ, (13)

where γ is made entirely of the type III cells of dimension m− 1. The torsion boundaryis the term 2γ , and γ is the torsion cycle. Formula (13) shows that the double cycle 2γ

is homologous to minus the free boundary of N . In the language of relative homology,γ is a relative torsion (m − 1)-cycle.

It is known that the torsion subgroup of Hm−1(N ,S;Z) is zero if N is orientable (seee.g. [Mas91, Bre93]). In that case, γ itself must be a relative boundary, say γ = ∂α−β

for some chains α ⊂ N and β ⊂ S. Then,

∂(N − 2α) = (∂N )free − 2β.

This shows that our cell decomposition had the wrong signs in the regions α and β.Reversing those signs provides a corrected cell decomposition that is more natural inthe sense that the ∂N calculated with its help does not introduce any spurious torsion

722 M. Alvarez

1

2

γ

B

B

1

2

2

1

1

2

C

C

s

s

s

s

Fig. 3. A cell decomposition of the Möbius band

boundaries. These are the “better” choices of orientation we mentioned earlier in relationto the cell decomposition of N .

The situation for non-orientable manifolds is different. No choice of cell decom-position can eliminate the torsion boundary, although it can change it by an absolutehomology. We are interested in the case where the frontier of the non-orientable mani-fold is orientable (the (m − 1)-cycle S). Then we can begin by orientating the frontier,that is, the type I (m − 1)-cells, and then provide the adjoining m-cells with matchingorientations. There is no natural prescription for orientating the rest of the cells.

These ideas are best illustrated by means of an example. The Möbius band is anon-orientable 2-manifold with boundary. It can be obtained from the 2-dimensionalprojective plane by cutting out an open disk, which leaves behind a one-dimensionalfrontier. A cell decomposition of the Möbius band that only uses two 2-cells is shownin Fig. 6. An orientation was first chosen for the frontier, and then the cells C1 and C2were orientated accordingly.

In the notation shown in the figure, the boundaries of the two cells are

∂C1 = s1 + s2 + B1 + γ,

∂C2 = −s1 − s2 + B2 + γ.(14)

This can be rewritten in terms of an incidence matrix as follows:

∂

(C1C2

)=

(1 1 1 0 1−1 −1 0 1 1

)⎛⎜⎜⎜⎝

s1s2B1B2γ

⎞⎟⎟⎟⎠ .

From the incidence matrix we see that B1 and B2 (columns 3 and 4) are type I cells,s1 and s2 (columns 1 and 2) are type II, and γ (column 5) is type III. Then, B1 and B2form a cell decomposition of the frontier of the Möbius band and γ is a torsion cycle.The boundary of the Möbius band is given by

∂N = ∂(C1 + C2) = B1 + B2 + 2γ, (15)

which is of the general form (13), with B1 + B2 the free boundary S, and 2γ the torsionboundary.

Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces 723

7. Stokes Theorem for Cell Decompositions

We are now in a position to apply the Stokes theorem for differential forms (6) to a closed(m−1)-form F that exists in a compact m-manifold N , not necessarily orientable, withnon-empty frontier S. We start by introducing a cell decomposition of the manifold.Because each m-cell has been provided with an orientation, (6) can be applied to it,

∫

V ma

d F =∫

∂V ma

F. (16)

Recalling that F is closed, this shows that∫

∂V ma

F = 0. (17)

Let us consider what will happen when the contribution of all the m-cells is added.Depending on whether the orientations of two adjoining m-cells agree or not, the (m−1)-cells that separate them will be type II or III. The orientations induced on a type II cell bythose of the two m-cells that coincide on it will be opposite. Therefore, F is integratedtwice on every type II cell, but with opposite orientations. The two integrals will cancelout, and so we find that type II cells make no contribution to the sum. In contrast, theinduced orientations will agree on all type III cells, so that the two fluxes will add upinstead of cancelling out.

Finally, the contribution of type I cells is straightforward: it is the flux of F on thefree boundary of N , that is, on its frontier S. In fact, the way the fluxes of F combinewhen all the contributions from the cells are added reproduces the calculation of thehomology boundary of N , as defined in (10),

∑a

∫

∂V ma

F =∫

∂NF.

When this is combined with (17), the result is what we regard as the Stokes theorem forclosed differential forms in a format that is applicable whether N is orientable or not,

∫

∂NF = 0. (18)

It has already been explained that, if N is orientable, then ∂N coincides with its frontierS, in which case (18) is a repetition of (7). On the other hand, if N is non-orientable,∂N consists of two distinct parts, namely the free and torsion boundaries, as in (13),and then

∫

SF = −2

∫

γ

F. (19)

The left-hand side of this is Φ(S), the F-charge contained inside S. The result (19)shows that the F-charge need not be zero, as we now discuss.

724 M. Alvarez

γF

SP

PQ

Q

Fig. 4. A torsion cycle acting as a magnetic source

8. Homological Aspects of F-Charge

The result (19) shows that, if N is non orientable, it may contain a non-zero F-charge,

Φ(S) = −2∫

γ

F. (20)

It is the presence of the torsion cycle γ that allows the flux of F through S to be nonzero.Because the torsion subgrop of Hm−1(N ,S;Z) is cyclic of order two, a γ will alwaysexist and be unique up to homology. In fact, (20) depends only on the absolute homologyclass of γ ,

∫

γ +∂C

F =∫

γ

F +∫

C

d F =∫

γ

F.

Therefore it is the absolute homology class [γ ]abs, rather than any particular choice ofγ , that acts as a magnetic source. In contrast with a magnetic current, which is localisedat the world-line of a magnetic charge, [γ ]abs is not a localised object, and for that reasonit is not meaningful to ask for a spacetime picture of how exactly the flux appears in Nand then crosses the sphere S. A rough representation of the process can be given if achoice of γ is made; Fig. 4 illustrates this point for a Möbius strip. In that figure, thefrontier of the Möbius strip is denoted S to match our notation for bounding spheres.

Let us recall that γ was originally a representative of Hm−1(N ,S;Z). Let [γ ]rel bethe relative homology class of γ in N . As explained, the F-charge (20) is invariant onlyunder absolute homologies of γ . Therefore the absolute homology class [γ ]abs should beunderstood as the preimage of [γ ]rel under the map j∗ in the homology exact sequenceof the pair (N ,S),

· · · i∗−→ Hm−1(N ;Z)j∗−→ Hm−1(N ,S;Z)

∂−→ Hm−2(S;Z)i∗−→ · · · .

For the exceptional case m = 2 we use reduced homology groups in dimension zero,and the sequence is still exact. Then the preimage of [γ ]rel in Hm−1(N ;Z) always existsfor m ≥ 2 because then Hm−2(S;Z) (or its reduced version) is zero.

Torsion Cycles as Non-Local Magnetic Sources in Non-Orientable Spaces 725

9. Existence of Manifolds Supporting Non-Zero F-Charge

The existence of a torsion (m − 1)-cycle γ is a necessary condition for the F-chargenot to be identically zero, but it is not sufficient. We also need the space manifold K tosupport closed (m − 1)-forms (such as F) with non-zero periods over γ . Thus we needto analyse the cohomology of the class of non-orientable manifolds that we have beenconsidering in this paper, and not just the homology.

A quick way to find the cohomology groups of a manifold given its homology groupsis to apply Poincaré duality [Mas]. Unfortunately, Poincaré duality for non-orientablemanifolds requires the use of Z2 coefficients, in which case F cannot be a differen-tial form, unless it is zero. For real (or integer) coefficients, there is no straightforwardrelationship between cohomology and homology of a non-orientable manifold.

We now proceed to show that the non-orientable manifolds we have been consideringdo in fact possess the correct cohomology structure to support non-zero F-charges, atleast when M is R

m minus a solid m-ball and S the boundary of that ball. Then Mhas the same de Rham cohomology as S, and in particular it admits a globally definedclosed (m − 1)-form with non-zero periods on S. That (m − 1)-form is a candidatefor the field strength F , bearing in mind that F should be globally defined in the entirespace manifold K and not just in M. Then we must find out whether this candidate forF , which we shall call FS , extends to N , the non-orientable piece of space boundedby S. The following argument, based on the Mayer-Vietoris exact sequence of the triad(X ;N ;B′), shows that it does and hence FS extends to a globally defined (m−1)-formF in K, as required.

Recall that N is constructed by removing an open m-ball B from some compact non-orientable m-manifold X without frontier. Let B′ be a slightly larger open ball so thatX = (interion N )∪ (interion B′). The Mayer-Vietoris sequence of (X ;N ;B′) containsthe section

· · · ← Hm(X )← Hm−1(N ∩ B′)← Hm−1(N )⊕ Hm−1(B′)← Hm−1(X )← · · ·(21)

(see for example [BT, Mas]). The subspace N ∩ B′ is a small neighbourhood of thesphere S so that Hm−1(N ∩ B′) = Hm−1(S). Then FS will fit into (21) as a represen-tative of Hm−1(S). Because FS is a differential form we need only consider the freeparts of the cohomology groups, and in particular we can put Hm(X) = 0. Noting alsothat Hm−1(B′) = 0, (21) reduces to

0← Hm−1(S)← Hm−1(N )← Hm−1(X )← · · · . (22)

Therefore every differential form in Hm−1(S) lifts to a differential form in N , and inparticular FS lifts to the form F that we identify as the field strength.

In conclusion, the non-orientable manifolds discussed in this paper possess the correcthomology and cohomology structure to support non-zero F-charges.

Before discussing examples, we shall show that the (m − 1)st relative homologygroup of N is isomorphic to the absolute homology group of X of the same dimension.Then the problem of identifying the torsion cycle in N is reduced to finding the torsioncycle in the original boundaryless manifold X . The argument is based on the relativehomology sequence of the triad (X ,N ,B′) [Mas91], which contains the section

Hm(X ,X ) −→ Hm−1(N ,S) −→ Hm−1(X ,B′) −→ Hm−1(X ,X ).

726 M. Alvarez

It follows that Hm−1(N ,S) ≈ Hm−1(X ,B′) because the first and last homology groupsshown are always zero. On the other hand the homology sequence of the pair (X ,B′)contains the section

Hm−1(B′) i∗−→ Hm−1(X )j∗−→ Hm−1(X ,B′) ∂−→ Hm−2(B′)

in which the first and last homology groups are zero (taking reduced homology in dimen-sion zero if m = 2). Then Hm−1(X ) ≈ Hm−1(X ,B′). In combination with the previousresult, this shows that Hm−1(N ,S) ≈ Hm−1(X ), as required.

The real projective plane RP2 and the Klein bottle K are the basic examples for

m = 2. When a 2-ball is removed from RP2 the result is a Möbius strip, an example that

has already been discussed.The first homology group of the Klein bottle with integer coefficients is Z⊕Z2, and

the cycle γ whose homology class acts as a non-local magnetic source after a ball isremoved from K is the generator of the torsion part Z2. This example is more genericthan the Möbius band in that the homology group contains both free and torsion parts,but the discussion in entirely analogous.

Easy examples with m = 3 are RP2×S1 and K×S1. The two-cycle whose homology

class acts as a magnetic source is now γ × S1, in which γ is the one-cycle in RP2 or K

described above. A less obvious example is S1×S2, the non-trivial S2 bundle over S1

[Ste51]. It is constructed by identifying the two spheres at the ends of S2 × I after anorientation-reversing transformation. Now H2(S1×S2) = Z2, pure torsion, generatedby a two-cycle that can be identified with the fibre S2 at any point in the base S1. Asusual, the absolute homology class of that cycle is the source of magnetic flux after athree-ball is removed from the manifold.

The manifold S1×S2 has been discussed previously in the literature [FM82] in rela-tion to the idea that sourceless electric charges can be defined in a non-orientable mani-fold. The work of [FM82] is based on a geometric construction that requires the manifoldto possess a rotational Killing pseudovector. An “electric field” is then defined in termsof covariant derivatives of the Killing vector and non-zero fluxes through a sphere arethen obtained, in the absence of electric charges. Our construction is purely topologicaland therefore independent of such considerations. Note also that our identification of thenon-zero charge as magnetic was fixed by the convention that the field strength (m−1)-form F was globally defined. The opposite convention, that the 2-form ∗F is globallydefined, would have yielded non-zero electric fluxes by an analogous construction, butonly in the particular case m = 3 where torsion cycles are 2-dimensional.

Acknowledgements. I would like to thank Rafael Sorkin, Paul Martin, Anton Cox, Mark Hadley and AllenHatcher for stimulating discussions on different aspects of this work, and especially Esperanza Gómez forher constant encouragement. This research was supported by PPARC through the Advanced FellowshipPPA/A/S/1999/00486.

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Communicated by G.W. Gibbons