Computing twisted bordism groups

For the classification problems we are interested in, this construction will always suffice.

The next step is to develop tools to calculate the bordism groups of twisted fibrations overBO.

3.4. Computing twisted bordism groups

We want to be able to use the methods of stable homotopy theory to calculate twisted bordism groups. Consequently, we need to construct spectra whose stable homotopy groups are isomorphic to the twisted bordism groups which we want to determine. The construction follows along the lines of Chapter 12 in [Swi02].

We obtain spectra for twisted bordism by modifying the construction of Thom spectra for BOhmi-bordism slightly. In order to distinguish Thom spaces from Thom spectra we denote the first by T h(.) and the latter by M(.). For any map f of vector bundles, we denote byT h(f) the induced map between the Thom spaces of the bundles.

To construct spectra as in Chapter 12 of [Swi02] we need (strictly) commutative diagrams BOhmin−1

on,m //

prn−1,m

BOhmin prn,m

BO_n−1 ^{on //}BO_n.

(5)

We obtain those by using a functorial construction for the (m−1)-connected cover of a simply connected space. Instead of working over BO we can always work over BSO since there exist commutative diagrams

BSO_n−1 ^//

BSO_n

BOn−1 //BOn,

(6)

where all maps are induced by the respective inclusions of the underlying groups.

By [Whi78], Chapter IX the Postnikov tower of a simply connected CW complex can be constructed functorially.

Applying the construction toBSO_n we obtain mapsBSO_n→Pm−1BSO_n. By the long exact sequence of homotopy groups of a fibration, the homotopy fibre of this map is the (m−1)-connected cover BOhmin of BSO_n. Since the homotopy fibre can also be constructed functorially we obtain commutative squares

BOhmin−1 on,m //

pren−1,m

BOhmin pren,m

BSO_n−1 ^{on,1 //}BSO_n.

(7)

3.4 Computing twisted bordism groups

Combining the commutative squares of diagram (6) and (7) we obtain the commutative square in diagram (5). Using the observations above, we can now sketch the construction of the Thom spectrum for BO- and BOhmi-bordism denoted by M O and M Ohmi, respectively.

Let γ_n^u →BO_n denote the universal vector bundle of rank nand let R → BO_n denote the trivial line bundle. There exist a bundle mapo_n:o^∗_nγ_n^u →γ_n^u covering o_n and there exists an isomorphismfn:γ_n−1^u ⊕R→o^∗_nγ_n^u. Composing both and passing to the Thom spaces we obtain a map

σn:=T h(on◦fn) :T h(γ_n−1^u )∧S¹ →T h(γ_n^u).

The spectrumM O consists of the spacesT h(γ_n) together with the maps σ_n.

Denote the pullback ofγ_n^ualongpr_n,m byγ_n,m^u . Completely analogously to the construc-tion above we obtain the spectraM Ohmi:

There exists a bundle mapo_n,m:o^∗_n,mγ_n,m^u →γ_n,m^u and, by commutativity of the diagram (5), a bundle isomorphism fe_n: (γ_n−1,m^u ⊕R)→o^∗_n,mγ_n,m^u .

By composing both maps and by passing to the Thom spaces we obtain maps σ_n,m:=T h(o_n,m◦fe_n) :T h(γ_n−1,m^u )∧S¹ →T h(γ_n,m^u ).

The spectrum M Ohmi consists of the Thom spaces T h(γ_n,m^u ) together with the maps σ_n,m.

Now we come to the construction for twisted bordism.

Recall that the fibrationX×eBO_n−r is defined by

X×BOhmi ^{ir◦E×pm //}BO×BO ^{⊕ //}BO ,

forE the classifying map of a vector bundle of rankroverX. We denote the total space of the bundle byE, too. There are commutative diagrams

X×BOhmin−1−r

1X×on−r,m //

E×prn−1−r,m

X×BOhmin−r E×prn−r,m

BOr×BOn−r−1 1BOr×on−r //

⊕

BOr×BOn−r

⊕

BO_n−1 ^{on //} BO_n. Thus, we obtain another sequence of Thom spaces

X_n:=T h(E×γ^u_n−r,m)∼=T h(E)∧T h(γ_n−r,m^u ).

3.4 Computing twisted bordism groups The maps1E ×on−r,m◦fen−r induce

1T h(E)∧σ_n−r,m:X_n−1∧S¹→X_n.

Let M(E×γ^u) be the spectrum consisting of the spaces X_n together with the maps 1T h(E)∧σ_n−r,m. By the Pontjagin-Thom construction we know

Ω^Ohmi_n (X, E)∼=π_n^st(M(E×γ^u)).

Thus, we can now apply the Adams spectral sequence to computeπ_n^st(M(E×γ^u)), i.e.

Ω^Ohmi_n (X, E)

In addition, it is helpful to have a modified version of an Atiyah-Hirzebruch spectral sequence which we introduce now. It is particularly helpful, because we can use it to determine which torsion can appear in Ω^Ohmi_n (X, E). For this purpose, we use it, e.g. in Sections 4.3 and 5.4.

We denote thet-fold suspension of a spectrumA by Σ^tA.

By constructionM(E×γ^u)'T h(E)∧Σ^−rM Ohmi 'Σ^−r(T h(E)∧M Ohmi). Hence, Ω^Ohmi_n (X, E) ∼= π_n^st(Σ^−r(T h(E)∧M Ohmi))

∼= π_n+r^st (T h(E)∧M Ohmi)∼= Ω^Ohmi_n+r (T h(E), pt).

Remark 3.15. On a geometric level the isomorphism T which is given by the compo-sition

Ω^Ohmi_n (X, E)→Ω^Ohmi_n+r (T h(E), pt)→Ω^Ohmi_n+r (D(E), S(E))

maps an element [M, f×α] to [(D(f^∗E), S(f^∗E)),fe×α]. Heree αeis theOhmi-structure on D(f^∗E), obtained by composing the projection of the disc bundle withα, and feis the bundle map coveringf.

To compute Ω^Ohmi∗ (T h(E), pt) we can use the usual Atiyah-Hirzebruch spectral sequence converging to the reduced ordinaryOhmi-bordism groups, i.e.

E_pq² =He_p(T h(E); Ω^Ohmi_q (pt))⇒Ω^Ohmi_p+q (T h(E), pt).

By the Thom isomorphism for oriented vector bundles

H_p(X; Ω^Ohmi_q (pt))∼=He_p+r(T h(E); Ω^Ohmi_q (pt)).

Since Ω^Ohmi_p+q (X, E)∼= Ω^Ohmi_p+q+r(T h(E), pt) there is a spectral sequence withE²-page E_pq² ∼=H_p(X; Ω^Ohmi_q (pt))

3.4 Computing twisted bordism groups converging to Ω^Ohmi_p+q (X, E).

We refer to this as thetwisted Atiyah-Hirzebruch spectral sequence.

Note, that the E²-page for the twisted Atiyah-Hirzebruch spectral sequence is exactly the same E²-page as the one of the Atiyah-Hirzebruch spectral sequence converging to the ordinary Ohmi-bordism group Ω^Ohmi_p+q (X). But even d₂-differentials can differ if w₂(E)6= 0, as we will see below.

Recall that, for m≥4, we have Ω^Ohmi₀ (pt)∼=Z and Ω^Ohmi_i (pt)∼=Z/2 for i= 1,2. Thus, entries in the 0-, 1- and 2-line on the (twisted) Atiyah-Hirzebruch spectral sequence are given by homology with coefficients in Zand Z/2, respectively.

In general we do not know the differentials in the (twisted) Atiyah-Hirzebruch spectral sequence. But we can say something about some of the d₂-differentials. The following lemma is due to [Tei93] for m = 4 but the proof follows completely analogously for m >4.

Lemma 3.16. Let X be a CW-complex andE→X a (possibly trivial) twisting bundle.

Consider the (twisted) Atiyah-Hirzebruch spectral sequence converging to Ω^Ohmi_n (X, E) for m≥4.

1. Let w₂:=w₂(E). The differentiald₂:E_p+2,1² →E_p,2² is dual to Sq²_w2:H^p(X;Z/2) → H^p+2(X;Z/2),

x 7→ Sq²(x) +x∪w₂.

2. Let Sq_w²₂ be defined as above. The differential d2:E²_p+2,0 → E_p,1² is given by the composition d₂ = (Sq²_w2)^∗ ◦red, where red: H_p+2(X;Z) → H_p+2(X;Z/2) is the reduction mod two.

Another helpful tool for computations is a sequence that allows us to compare twisted Ohli-bordism ofCP^m andCP^m−1. We start with the definition of a map which appears within the sequence but also in a more general setting.

Letξ→X be a real bundle over a manifoldX which contains a codimensionr subman-ifold Y ⊂X and letξ⁰ denote the normal bundle of Y ,→X.

Definition 3.17. Let M be a smooth, closed n-dimensional manifold and f:M → X a map such that νM ⊕f^∗(−ξ) admits a Ohli-structure α:M → BOhli. Consider the induced element [M, f×α]∈Ω^Ohlin (X, ξ), We can always assume that f |∩Y. LetN be the preimagef⁻¹(Y). Since

ν(N)∼=ν(N ,→M)⊕ν(M)|N ∼= (f|N)^∗ξ⁰⊕ν(M)|N, the Whitney sum of ν(N) and (f|N)^∗(−(ξ|Y ⊕ξ⁰)) fulfills

ν(N)⊕(f|N)^∗(−(ξ|Y ⊕ξ⁰))∼=ν(M)|N ⊕(f|N)^∗(−ξ|Y).

3.4 Computing twisted bordism groups Consequentlyα|N defines aOhli-structure twisted by ξ|Y ⊕ξ⁰. Thus, we can define

t: Ω^Ohli_n (X, ξ)→Ω^Ohli_n−r(Y, ξ|Y ⊕ξ⁰), by [M, f×α]7→[N,(f ×α)|N].

This map is well-defined by the following observation. Given two representatives of [M, f ×α]∈ Ω^Ohli_n (X, ξ) they are, by definition, bordant in Ω^Ohli_n (X, ξ). We apply the construction in the definition of t to the bordism and obtain that the images of the representatives undertare bordant in Ω^Ohli_n−r(Y, ξ|Y ⊕ξ⁰).

Now we come to the special case of X = CP^m and Y = CP^m−1. We obtain a long exact sequence relating twisted bordism ofCP^m andCP^m−1 which is well-known to the experts. Applications of a similar sequence can be found in [Kre09]. But there does not seem to be a published proof. Therefore, we also give a proof.

Lemma 3.18. Let ξ → CP^m be an oriented real vector bundle of finite rank. Let i: Ω^Ohli_k (pt) → Ω^Ohli_k (CP^m, ξ) be the map induced by the inclusion of a point into CP^m and let H := ν(CP^m−1 → CP^m) denote the Hopf bundle. Then the following sequence is exact:

...→Ω^Ohli_n (pt)→ⁱ Ω^Ohli_n (CP^m, ξ)→^t Ω^Ohli_n−2(CP^m−1, ξ|_CP^m−1 ⊕H)→^s Ω^Ohli_n−1(pt)→... . The maps will be constructed in the proof.

To prove this lemma we need the following easy observation.

Lemma 3.19. Let ξ → CP^m be a real vector bundle, M a compact, smooth manifold and f: M → CP^m−1 a map. Furthermore, let p_S: S(f^∗H) → M denote the sphere bundle off^∗H.

Then the bundle p^∗_Sf^∗(ξ|_CP^m−1) is trivial.

Proof. Lete^mdenote the top cell ofCP^m. We obtain the following commutative diagram of total spaces

S(f^∗(H))_{ _} ^//

pS

""

S(H)_{ _}

 //e^m_{ _}

D(f^∗(H)) ^{f //}

D(H)

 //CP^m

M ^{f //}CP^m−1

Recall that CP^m =D(H)∪S(H)e^m, i.e. S(H) bounds the top disc e^m. Every bundleξ overCP^m becomes trivial under restriction toe^m and thus, under restriction to S(H).

By the commutativity of the diagram the pullbackp^∗_Sf^∗ξ|_CP^m−1 is also trivial.

3.4 Computing twisted bordism groups

Now we are ready to prove Lemma 3.18. Within the proof we suppress the decoration Ohli in the notation of the twisted bordism groups.

Proof. We start with the definition of s: Ω_n−2(CP^m−1, ξ|_CP^m−1 ⊕H)→Ω_n−1(pt). We claim that

s: Ω_n−2(CP^m−1, ξ|_CP^m−1 ⊕H) → Ω_n−1(pt) defined by [M, f×α] 7→ [S(f^∗H), pt×(α◦p_S)],

is a well-defined map. Herept denotes the constant map to a point. We need to check that [S(f^∗H), pt×α◦pS] is an element in Ωn−1(pt), i.e. we need to show thatα◦pS is an Ohli-structure on the total space of the sphere bundlep_S:S(f^∗H)→M.

The stable tangent bundle ofS(f^∗H) is isomorphic top^∗_ST M⊕p^∗_S(f^∗H), wherep^∗_S(f^∗H) is trivial by Lemma 3.19 since we can consider the Hopf bundle overCP^m−1as restriction of the Hopf bundle over CP^m. Thus, the stable normal bundle ν(S(f^∗H)) is isomor-phic to p^∗_S(ν(M)). By Lemma 3.19 the bundle p^∗_Sf^∗(−(H⊕ξ)|_CP^m−1) is also trivial.

Consequently, we obtain

ν(S(f^∗H))∼=p^∗_S ν(M)⊕f^∗(−(H⊕ξ)|_CP^m−1) .

By assumption,αis anOhli-structure onν(M)⊕f^∗(−(H⊕ξ)|_CP^m−1). Hence, the com-positionα◦p_S is aOhli-structure on the normal bundle ν(S(f^∗H)).

Well-definedness of s follows, again, by applying the construction of s to a twisted CP^m−1×eBOhli-bordism between two representatives of [M, f ×α].

Now, we prove the exactness. We start by showing that im(i)⊂ker(t).

Let [M, pt×α]∈ Ω_n(pt). The map pt:M → CP^m is transversal to CP^m−1 ⊂ CP^m if pt /∈CP^m−1. Thus, by definition of t, the compositiont◦ivanishes.

Next, we show ker(t)⊂im(i).

Let [M, f×α]∈ker(t)⊂Ω_n(CP^m, ξ) and let [N,(f ×α)|N] :=t([M, f ×α]). Since, by assumption, [N,(f×α)|N] = 0 there exists aCP^m−1×eBOhlizero-bordismW, i.e. ∂W = N and there exists a mapF×β:W →CP^m−1×BOhlisuch that (F×β)|N = (f×α)|N. By definition, β is an Ohli-structure on the Whitney sumν(W)⊕F^∗(−(ξ⊕H)).

Let D denote the total space of the disc bundle p_D:D(F^∗H) → W. Since the normal bundle of D is isomorphic to p^∗_D(ν(W)⊕F^∗(−H)), β◦pD is an Ohli-structure on the sum ν(D)⊕p^∗_DF^∗(−ξ). Furthermore, there are isomorphisms D|N ∼= D((f|N)^∗H) ∼= Dν(N ,→M) =:D⁰. In particular, there is an embedding D⁰,→M× {1}. Thus, we can construct a bordism

W⁰ :=M ×I∪D⁰D.

It admits a twistedOhli-structure and a map toCP^mwhose restriction toM×{0} ⊂W⁰ isf ×α.

3.4 Computing twisted bordism groups

The other boundary component ofW⁰isM⁰= (M−D⁰)∪S(F^∗H). By construction there is a map F: S(F^∗H) → S(H) ⊂ e^m, covering F, which is homotopic to the constant map. By Lemma 3.19β◦p_S is anOhli-structure onν(S(F^∗H)).

The mapf|M−D⁰ is also homotopic to the constant map since im(f|M−D⁰)⊂e^m⊂CP^m. Thus, the restrictionf^∗(−ξ)|M−D⁰ is trivial. Consequently,α|M−D⁰ is an Ohli-structure on ν(M−D⁰) =ν(M)|M−D⁰. Hence, we obtain an element [M⁰, pt×(α|M−D⁰ ∪β◦p_S)]

in Ωn(pt) whose image under iis, byW⁰, bordant to [M, f×α]∈Ωn(CP^m, ξ).

We proceed by showing that im(t)⊂ker(s).

Let [M, f ×α]∈Ω_n(CP^m, ξ) and let [N,(f ×α)|N] :=t([M, f ×α]). We need to show thatS :=S((f|N)^∗H) is zero-bordant in Ω_n−1(pt).

Note thatS ∼=S(ν(N ,→ M)). Thus, it is the boundary of W :=M −D(ν(N ,→M)).

By construction we again obtain that im(f|W)⊂e^m⊂CP^m and thus,f|W is homotopic to the constant map implying thatf^∗(−ξ)|W is trivial. Consequently, the restrictionα|W

is anOhli-structure onν(M)|W ∼=ν(W). Hence, W is a zero-bordism ofS in Ωn−1(pt).

Of course, the next step is to show that ker(s)⊂im(t).

Now assume that [N, g×β] is in the kernel of s, i.e. [S(g^∗(H)), pt×(p_S◦β)] is zero-bordant by some bordism W withOhli-structure α which restricts to p_S◦β.

Let D denote the total space of the disc bundle p_D:D(g^∗H) → N. Its normal bundle isν(D)∼=p^∗_D(ν(N)⊕g^∗(−H)). Thus, β◦p_D is anOhli-structure onν(D)⊕p^∗_Dg^∗(−ξ).

Furthermore, there is a bundle mapg:D(g^∗H)→D(H),→CP^m coveringg. As before, the image of g|S(g^∗H) is contained in the the top cell e^m ⊂ CP^m, i.e. g|S(g^∗H) ' pt.

Consider M := W ∪S(g^∗H)D. It admits a map f := pt∪g:M → CP^m. Furthermore, α∪β◦p_D is anOhli-structure onν(M)⊕f^∗(−ξ). By constructiont([M, f×(α∪p◦β)]) = [N, g×β].

It remains to show that im(s) = ker(i).

Let [M, f ×α] ∈ Ω_n−2(CP^m−1,(ξ⊕H)|CP^m−1). We need to construct a zero-bordism of i◦s([M, f ×α]) = i([S(f^∗H), pt×α]) in Ω_n−1(CP^m, ξ). Consider the disc bundle with total space D := D(f^∗H) and projection p_D, together with the covering map f:D(f^∗H) → D(H) ⊂ CP^m. Since ν(D) ∼= ν(M)⊕p^∗_Df^∗(−H), α◦p_D is an Ohli -structure onν(D)⊕p^∗_Df^∗(−ξ). Thus,Dtogether withf andα◦p_D is our zero-bordism.

Finally, we show that ker(i)⊂im(s).

For this purpose, let [M, pt ×α] ∈ ker(i) ⊂ Ωn−1(pt). Let W be a zero-bordism of i([M, pt ×α]) ∈ Ω_n−1(CP^m, ξ), i.e. there exists F: W → CP^m such that ν(W)⊕ F^∗(−ξ) admits an Ohli-structure β which restricts to α. Assume that F is transversal toCP^m−1 ⊂CP^mand letN :=F⁻¹(CP^m−1) andF|N =:f. Thenν(N)∼=ν(W)⊕f^∗H andβ|N is an Ohli-structure forν(N)⊕f^∗(−(ξ⊕H)). Consequently, [N, f×β|N] is an element in Ω_n−2(CP^m−1, ξ⊕H).

Consider W⁰ := W −D(f^∗H). Then β restricts to an Ohli-structure on ν(W⁰) since

3.4 Computing twisted bordism groups

im(F|W⁰) ⊂ e^m ⊂ CP^m, i.e. (F|W⁰)^∗(−ξ) is trivial. Thus, W⁰ is a bordism between s([N, f×β|N]) and [M, pt×α].

Eight-dimensional cohomology Bott manifolds

Im Dokument On the Classification of Cohomology Bott Manifolds (Seite 31-39)