Non-orientable Poincare Duality

This post is based on Differential Forms on Algebraic Topology written by Raoul Bott and Loring W. Tu. Without specifically emphasized, the coefficient ring of the cohomology is R\mathbb{R}, and the cohomology ring is defined to be the ring of differential forms and wedge product. The nondegenerate pair, as appears in Algebraic Topology by Allen Hatcher called cap product \cap, is now the integration operation along (sub)manifolds.

The purpose of this article is to understand the Poincare Duality in the non-orientable manifolds. The mathematical objects that we are using is the twisted de Rham complex.

Twisted de Rham complex

First we consider a simple case, when VV is a vector space.

Def. (Twisted differential forms) Let MM be a manifold and VV be a vector space. The space of twisted differential forms with values in VV, denoted by Ω(M,V)\Omega^*(M,V) is spanned by

ωv\omega\otimes v

where wΩ(M)w\in\Omega^*(M) and vVv\in V. The differential operator dd is defined to be the linear expansion of

d(ωv)=(dω)v.d(\omega\otimes v)=(d\omega)\otimes v.

Remark. With the above differential, the cohomology ring of Ω(M,V)\Omega^*(M,V) can be defined. If VV is the vector space of dimension nn, then


To define the notion of twisted in a more general case, i.e. when EE is a vector bundle, we shall know how to differentiate the “vectors” in EE. Think about “vectors in EE” being “sections”, and the differentials can be defined through the process of local trivialization of EE.

Def. (EE-valued qq-forms) Let EE be a vector bundle. Define the space of EE-valued qq-forms Ωq(M,E)\Omega^q(M,E) being expanded by global sections of (ΛqTM)E(\Lambda^qT^*M)\otimes E, i.e.

s:M(qTM)E.s:M\rightarrow \left(\wedge^qT^*M\right)\otimes E.

Suppose that {(Uα,ϕα)}\{(U_\alpha,\phi_\alpha)\} is a local trivialization of EE, where

ϕα:EαUα×Rk.\phi_\alpha:E_{\alpha}\rightarrow U_\alpha\times\mathbb{R}^k.

Let eα1,,eαke_\alpha^1,\ldots,e_\alpha^k be the sections of EαE_\alpha corresponding to the standard basis of Uα×RkU_\alpha\times\mathbb{R}^k, called a standard locally constant sections. ωΩq(M,E)\forall \omega\in\Omega^q(M,E), it has a local expression on UαU_\alpha

ω=iωieαi,\omega=\sum_i\omega_i\otimes e_\alpha^i,

where ωiΩq(M)\omega_i\in\Omega^q(M). With the local expression, define the differential on UαU_\alpha to be

dω=i(dωi)eαi.d\omega=\sum_i(d\omega_i)\otimes e_\alpha^i.

Compatibility: Suppose on the overlapping set UαβU_{\alpha\beta},

ω=ωieαi=τjeβj.\omega=\sum\omega_i\otimes e_\alpha^i=\sum\tau_j\otimes e_\beta^j.

where eαi=jcijeβje_\alpha^i=\sum_jc_{ij}e_\beta^j. Then

i(dωi)eαi=i,jcijdωieβj=jd(icijωi)eβj=j(dτj)eβj.\begin{aligned} \sum_i (d\omega_i)\otimes e_\alpha^i &=\sum_{i,j}c_{ij}d\omega_i\otimes e_\beta^j\\ &=\sum_{j}d(\sum_i c_{ij}\omega_{i})\otimes e_\beta^j\\ &=\sum_j (d\tau_j)\otimes e_\beta^j. \end{aligned}

Hence dωd\omega is globally defined on MM. Therefore, the above procedure defines a cohomology group of EE-valued forms Hϕ(M,E)H_\phi^*(M,E), where the subscript cannot be ignored as the cohomology might not be independent of the choice of local trivialization {(Uα,ϕα)}\{(U_\alpha,\phi_\alpha)\}.

e.g. Suppose that M=S1M=S^1 and E=S1×RE=S^1\times\mathbb{R}. Consider two local trivialization of EE being

  1. U=S1U=S^1, ϕ(x,h)h\phi(x,h)\rightarrow h;
  2. U1,U2U_1,U_2 where ϕ1:U1×RR\phi_1:U_1\times\mathbb{R}\rightarrow\mathbb{R} is


and ϕ2:U2×RR\phi_2:U_2\times\mathbb{R}\rightarrow\mathbb{R} is


for some (nowhere zero) function ρ\rho. In the first local trivialization, Hϕ0(M,E)=ker(d:Ω0EΩ1E)=ker(d:Ω0Ω1)=RH_\phi^0(M,E)=\ker(d:\Omega^0\otimes E\rightarrow\Omega^1\otimes E)=\ker(d:\Omega^0\rightarrow\Omega^1)=\mathbb{R}.

In the second local trivialization, the locally constant section ei:Ui(0TM)E=MEe_i:U_i\rightarrow\left(\wedge^0 T^*M\right)\otimes E=M\otimes E satisfies

  • e0(x)=(x,1)e_0(x)=(x,1),
  • e1(x)=(x,1/ρ(x))e_1(x)=(x,1/\rho(x)): the second coordinate is because ϕ2e1(x)=(x,ρ(x))\phi_2\circ e_1(x)=(x,\rho(x)*) should be locally constant near xx, where e1(x)=(x,)e_1(x)=(x,*).

Observe that a global EE-valued qq-form ω\omega cannot be written as the “gluing” of the locally constant form on U1U_1 and U2U_2. Therefore Hψ0(M,E)=0H_\psi^0(M,E)=0.

Cohomology with respect to local trivialization

Question: when will two local trivializations have the same cohomology group Hϕ(M,E)H^*_\phi(M,E)?

Prop. The twisted cohomology is invariant under the refinement of open covers.

Proof. The definition of differential operator is local. Thus if UαU_\alpha is a refinement of VβV_\beta, then the two local homeomorphism ϕ\phi and ψ\psi agrees on UαU_\alpha. Thus the differential operators coincide, and Ωϕ(M,E)=Ωψ(M,E)\Omega_\phi^*(M,E)=\Omega^*_\psi(M,E). \square

Say that two local trivialization {(Uα,ϕα)}\{(U_\alpha,\phi_\alpha)\} and {(Uα,ψα)}\{(U_\alpha,\psi_\alpha)\} differ by a locally constant comparison 00-cochain, if aαij:UαGL(k,R)\exists a_\alpha^{ij}:U_\alpha\rightarrow \mathrm{GL}(k,\mathbb{R}), s.t.

eαi=jaαijfαj.e_\alpha^i=\sum_j a_{\alpha}^{ij}f_\alpha^j.

Then there is an isomorphism

F:Ωϕq(M,E)Ωψq(M,E)eαijaαijfαjF:\Omega_\phi^q(M,E)\rightarrow\Omega_\psi^q(M,E)\\ e_\alpha^i\mapsto\sum_j a_\alpha^{ij}f_\alpha^j

which induces an isomorphism in cohomology F:Hϕq(M,E)Hψq(M,E)F^*:H^q_\phi(M,E)\rightarrow H_\psi^q(M,E). Therefore, if two local trivialization differs by a locally constant comparison 00-cochain, then they have the same cohomology. By taking a common refinement, we can always compare two local trivialization on the same open cover {Uα}\{U_\alpha\}.

Prop. Let EE be a flat vector bundle of rank kk and {gαβ}\{g_{\alpha\beta}\} and {hαβ}\{h_{\alpha\beta}\} be transition functions for EE relative to two locally constant trivialization ϕ\phi and ψ\psi for the same open cover. If there exists a locally constant functions


such that gαβ=λαhαβλβ1g_{\alpha\beta}=\lambda_\alpha h_{\alpha\beta}\lambda_{\beta}^{-1}, then there are isomorphisms



Hϕ(M,E)Hψ(M,E).H^*_\phi(M,E)\simeq H_\psi^*(M,E).

Orientation line bundle

To understand whether an manifold is orientable, we can consider its orientation bundle.

Def. (Orientation bundle) Let gαβ:=ϕαϕβ1g_{\alpha\beta}:=\phi_\alpha\circ\phi_{\beta}^{-1} be the transition map. Define the orientation bundle of MM to be the line bundle LL determined by

sgnJ(gαβ)\mathrm{sgn} J(g_{\alpha\beta})

Remark. MM is orientable iff the LL is trivial.