Orientation and incidence number

Cellular Homology

2.5 Orientation and incidence number

2. In the definition of “suspension” prove that Σg is continuous when gis; and thatΣg₁ andΣg₂ are homotopic wheng₁ andg₂ are.

3. Prove that homeomorphic CW complexes have the same dimension.

4. Describe a CW complex structure forRⁿ.

5. (for those who know category theory) Show that “wedge” is the category theoret- ical coproduct in the category Pointed Spaces. What is the category theoretical coproduct in the category Spaces?

2.5 Orientation and incidence number 53 Proof. First, we observe thatr|:Sⁿ⁻¹→Sⁿ⁻¹is not homotopic to id. Forn= 1, this is obvious. Forn >1, consider the homeomorphismt12:Sⁿ⁻¹→Sⁿ⁻¹ (x1, x2, x3, . . . , xn)→(x2, x1, x3, . . . , xn). By 2.4.3,r|=t⁻₁₂¹◦fn−1,−1◦t12. If r|were homotopic to id, thenfn−1,−1would be homotopic to id (conjugating the homotopy byt12), which is false by 2.4.4.

It follows that r : (Bⁿ, Sⁿ⁻¹) → (Bⁿ, Sⁿ⁻¹) is not equivalent to (i.e., pairwise homotopic to) the identity map. Clearly (h◦r):Bⁿ/Sⁿ⁻¹→eⁿ_α/^•eⁿ_α is the compositionh◦r. Sokn◦(h)⁻¹◦(h◦r)◦k_n⁻¹=kn◦r◦k_n⁻¹, and this map does not have degree 1. Thushandh◦rdetermine different orientations ofeⁿ_α. By 2.5.1, every orientation is given by one of these.

The proof of 2.5.1 depends on some lemmas.

Lemma 2.5.3.Let n≥1. Leta= (0, . . . ,0,1)∈Sⁿ andb= (0, . . . ,0,−1)∈ Sⁿ. Let eⁿ₊ and eⁿ₋ be the closed upper and lower hemispheres, respectively.

Letf, g:Sⁿ→Sⁿ be maps such thata∈f(eⁿ₋)∪g(eⁿ₋)andb∈f(eⁿ₊)∪g(eⁿ₊).

If f andg have the same degree, thenf| g|:Sⁿ⁻¹→Sⁿ− {a, b}.

Proof. We considerf andgas maps (Sⁿ, Sⁿ⁻¹)→(Sⁿ, Sⁿ−{a, b}). As in the first part of the proof of 2.4.18,f andg are (pairwise) homotopic to mapsf andg which preserve hemispheres, hence, as in the proof of 2.4.5, to suspen- sionsΣf andΣg, where f, g:Sⁿ⁻¹→Sⁿ⁻¹. These homotopies restrict to homotopiesSⁿ⁻¹×I→Sⁿ− {a, b}. Sof| inclusion◦fandg| inclu- sion◦ g (as mapsSⁿ⁻¹ →Sⁿ− {a, b}). Now degree(f) = degree(Σf) = degree(f) = degree(g) = degree(Σg) = degree(g). So f g, by 2.4.5, hencef| g|. Note that, while 2.4.18 is false forn= 1, the part of the proof

used here works forn= 1.

Corollary 2.5.4.Let f, g : (Bⁿ, Sⁿ⁻¹) → (eⁿ_α,^•eⁿ_α) be characteristic maps inducing f, g : Bⁿ/Sⁿ⁻¹ → eⁿ_α/^•eⁿ_α. For t ∈ (0,1), let A_t be the annulus {x ∈ Bⁿ | t < |x| < 1} and let S_t = {x ∈ Bⁿ | |x| = t}. Let z ∈ e^◦n_α. If there exists t such that f(At)∪g(At) ⊂ e^◦n_α− {z}, and if f g, then f | g|:S_t→eⁿ_α− {z}.

Proof. SinceBⁿ/Sⁿ⁻¹andeⁿ_α/^•eⁿ_αare homeomorphic toSⁿ, the result follows from 2.5.3. In fact,f | g|as maps fromSttoe^◦n_α− {z}. Lemma 2.5.5.Letz∈e^◦n_α, letf andgbe two maps(Sⁿ⁻¹×I, Sⁿ⁻¹×{1})→ (eⁿ_α− {z},e^•n_α), and let H :Sⁿ⁻¹× {0} ×I →eⁿ_α− {z} be a homotopy from f |Sⁿ⁻¹× {0} tog|Sⁿ⁻¹× {0}. Then there is a homotopy K: (Sⁿ⁻¹×I× I, Sⁿ⁻¹× {1} ×I)→(eⁿ_α− {z},e^•n_α)from f tog extendingH.

Proof. The map (Sⁿ⁻¹× {0} ×I)∪(Sⁿ⁻¹×I× {0,1})→eⁿ_α− {z}, which agrees with H onSⁿ⁻¹× {0} ×I, withf on Sⁿ⁻¹×I× {0} and withg on Sⁿ⁻¹×I× {1}, extends, by 1.3.15, to a mapL:Sⁿ⁻¹×I×I →eⁿ_α− {z}.

The sete^◦n_α−(image ofL) =:U is a neighborhood ofzin ^◦eⁿ_α. By 2.4.14, there is a homotopy D : X ×I → X such that D0 = id, Dt = id on X −^◦eⁿ_α for all t, and D1(eⁿ_α−U) ⊂ e^•n. Define M : Sⁿ⁻¹×I×I → eⁿ_α− {z} by M(x, s, t) = L(x, s,2t) if 0≤ t ≤ ¹₂ and by M(x, s, t) = D2t−1(L(x, s,1)) if

1

2 ≤t≤1. It is a matter of planar geometry to construct a homeomorphism h : I × I → I × Iagreeing with id onI× {0}, taking (0, t) to (0,2t) and (1, t) to (1,2t) when 0≤t≤ ¹₂, and taking the rest of fr_R2(I×I) intoI× {1}.

The requiredK isM◦h⁻¹.

Proof (of 2.5.1). “Only if” is obvious. We prove “if”. By the hypothesis and 2.4.5h₁ h₂ :Bⁿ/Sⁿ⁻¹→ eⁿ_α/^•eⁿ_α. Letz ∈e^◦n_α. There exists t∈(0,1) such that At is mapped into e^◦n_α− {z} by h1 and by h2. By 2.5.4, there exists H :St×I→eⁿ_α− {z}such thatH0=h1 |St andH1=h2 |St. Therefore,

by 2.5.5 and 1.3.17,h1h2.

An oriented CW complex is a CW complexX together with a choice of orientation on each cell. If A is a subcomplex of X, it is understood to be given the inherited orientation (each cell oriented as in X): we call (X, A) anoriented CW pair. A quotient complexX/∼, as in Sect. 1.2, is given the quotient orientation: the vertices {A_α} are oriented by +1; all other cells receive their orientation via the quotient mapX→X/∼.

LetX be an oriented CW complex,eⁿ_α ann-cell andeⁿ_β⁻¹ an (n−1)-cell.

First, assumen≥2. Choose characteristic mapshα: (Bⁿ, Sⁿ⁻¹)→(eⁿ_α,e^•n_α) andhβ: (Bⁿ⁻¹, Sⁿ⁻²)→(eⁿ_β⁻¹,^•eⁿ_β⁻¹) representing the orientations. Consider the commutative diagram:

Xⁿ⁻¹

oo ? _eⁿ⁻¹_β

oo ^hβ Bⁿ⁻¹

•

eⁿ_α

inclusion

77o

oo oo oo oo oo oo oo

s_αβ //Xⁿ⁻¹/(Xⁿ⁻¹−^◦eⁿ_β⁻¹)oo

r_β eⁿ_β⁻¹/^•eⁿ_β⁻¹oo

h_β Bⁿ⁻¹/Sⁿ⁻²

k_n−1

Sⁿ⁻¹

h_α|

OO

Sⁿ⁻¹

Here rβ is induced by inclusion and is clearly a homeomorphism. Unmarked arrows are quotient maps. sαβ is the indicated composition, and kn−1 is as before. Sinceh_β is a homeomorphism, we obtain a map

kn−1◦(h_β)⁻¹◦r⁻_β¹◦sαβ◦hα|:Sⁿ⁻¹−→Sⁿ⁻¹

whose degree, denoted by [eⁿ_α:eⁿ_β⁻¹], is theincidence number of the oriented cells eⁿ_αand eⁿ_β⁻¹. One should think of it as the (algebraic) number of times eⁿ_αis attached to eⁿ_β⁻¹. By 2.5.1 we have:

2.5 Orientation and incidence number 55 Proposition 2.5.6.For n ≥2, the definition of [eⁿ_α : eⁿ_β⁻¹] depends on the orientations but not on the specific mapshα andhβ. Note that the definition of [eⁿ_α :eⁿ_β⁻¹] also depends on k_n−1. Indeed, the homeomorphisms{k_n|n≥1}have been chosen once and for all. We will say more about how we want them to have been chosen later (see 2.5.16).

We now extend the definition of incidence number to the case n = 1.

Given an oriented 1-cell e¹_α and a 0-cell e⁰_β oriented by ∈ {±1}, choose a characteristic map hα : (B¹, S⁰) → (e¹_α,^•e¹_α) representing the orientation.

Define theincidence number [e¹_α:e⁰_β] by:

[e¹_α:e⁰_β] =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ifh_α(1) =e⁰_β and h_α(−1)=e⁰_β

− ifhα(1)=e⁰_β and hα(−1) =e⁰_β 0 ifh_α(1) =h_α(−1) =e⁰_β

0 ifhα(1)=e⁰_β and hα(−1)=e⁰_β. Clearly we have:

Proposition 2.5.7.The definition of [e¹_α :e⁰_β] depends only on the orienta-

tions.

Proposition 2.5.8.Leteⁿ_α andeⁿ_β⁻¹be (oriented) cells,n≥1. Ifeⁿ_β⁻¹ is not a subset of eⁿ_α, then[eⁿ_α:eⁿ_β⁻¹] = 0.

Proof. Whenn≥2, the mapSⁿ⁻¹→Sⁿ⁻¹ whose degree defines [eⁿ_α :eⁿ_β⁻¹] is not surjective, so, by 1.3.4 and 1.3.7, it is homotopic to a constant map. By 2.4.3 its degree is 0. Whenn= 1, the Proposition is obvious.

We sayeβ is aface ofeαifeβ⊂eα.

Now we turn to numbers associated with maps. LetX andY be oriented CW complexes, and letf :X →Y be a map such that, for somen,f(Xⁿ)⊂ Yⁿ andf(Xⁿ⁻¹)⊂Yⁿ⁻¹; of course cellular maps have this property for all n. For eacheⁿ_α in X and ˜eⁿ_β in Y we wish to define an integer [eⁿ_α : ˜eⁿ_β : f] measuring the (algebraic) number of timesf mapseⁿ_αonto ˜eⁿ_β. First, letn≥1 and consider the commutative diagram:

B^{n hα} //

eⁿ_α ^f| //

Yⁿ oo ? _

˜

eⁿ_β oo ^˜hβ

Bⁿ

Bⁿ/Sⁿ⁻¹

h_α

//

k_n

eⁿ_α/^•eⁿ_α

f_αβ

//Yⁿ/(Yⁿ−^◦˜eⁿ_β)oo

r_β ˜eⁿ_β/e^•˜ⁿ_βoo

˜h_β Bⁿ/Sⁿ⁻¹

k_n

Sⁿ Sⁿ.

Here ˜eⁿ_β is a cell ofY; ˜hβis a characteristic map defining the orientation;f_αβ is induced byf, and is well defined sincef(Xⁿ, Xⁿ⁻¹)⊂(Yⁿ, Yⁿ⁻¹);rβ is a homeomorphism as before. Let [eⁿ_α: ˜eⁿ_β :f] be the degree of the map

kn◦(˜h_β)⁻¹◦r⁻_β¹◦f_αβ ◦h_α◦k_n⁻¹:Sⁿ→Sⁿ.

For the case n = 0, let e⁰_α ⊂ X have orientation and let ˜e⁰_β ⊂ Y have orientation ˜. Define [e⁰_α : ˜e⁰_β : f] = ˜ iff(e⁰_α) = ˜e⁰_β, and [e⁰_α : ˜e⁰_β : f] = 0 otherwise. From 2.5.1 we conclude:

Proposition 2.5.9.The definition of[eⁿ_α: ˜eⁿ_β:f]depends only on the orien-

tations.

As with 2.5.8 we have:

Proposition 2.5.10.Let eⁿ_α and e˜ⁿ_β be oriented n-cells of X and Y respec- tively. If˜eⁿ_β is not a subset off(eⁿ_α)then[eⁿ_α: ˜eⁿ_β:f] = 0.

The number [eⁿ_α: ˜eⁿ_β :f] is called themapping degree off with respect to eⁿ_αand ˜eⁿ_β.

Next, we discuss product orientations. LetX andY be oriented CW com- plexes. We define the product orientation on the CW complex X ×Y by specifying an orientation for eache^m_α ×e˜ⁿ_β wheree^m_α, ˜eⁿ_βare cells of X andY respectively: (i) if m >0 and n > 0, and if hα : (I^m,I^•m) → (e^m_α,e^•m_α) and

˜hβ : (Iⁿ,^•Iⁿ)→(˜eⁿ_β,˜e^•n_β) are characteristic maps defining the separate orien- tations,e^m_α טeⁿ_β is to be oriented byh_αטh_β. (ii) Ifm >0 andn= 0, so that

˜

e⁰_β={y_β}, ifh_αorientse^m_α, and if 1 orients ˜e⁰_β, thene^m_α טe⁰_βis to be oriented by the mapI^m → e^m_α טe⁰_β, x→ (h_α(x), y_β); if−1 orients ˜e⁰_β, the opposite orientation is to be used. (iii) If m= 0 andn >0, the rule of orientation is similar to that given in (ii). (iv) Ifm= 0 =n, withe⁰_α oriented by, and ˜e⁰_β oriented by ˜, then e⁰_α×e˜⁰_β is to be oriented by˜.

We can use this to define a preferred orientation on the CW complexIⁿ as follows: whenn= 0, thecanonical orientation is 1; whenn= 1, thecanonical orientation onI¹(with two vertices and one 1-cell) is: 1 on each vertex, and (the equivalence class of) id :I¹→I¹on the 1-cell. Thecanonical orientation onIⁿ is then defined inductively to be the product orientation onIⁿ⁻¹×I¹. We leave the proofs of the next two statements to the reader:

Proposition 2.5.11.If Iⁿ is factorized as I^r×I^s, and each factor carries the canonical orientation, then the product orientation on I^r×I^s coincides

with the canonical orientation onIⁿ.

Proposition 2.5.12.Let r ≥ 1. Let e be the r-cell of Iⁿ obtained by con- straining the i_j^th coordinate to be _j ∈ {1,−1}, for 1 ≤ j ≤ n−r. Let the remaining r coordinates be indexed by p1 < p2 < . . . < pr. The canonical

2.5 Orientation and incidence number 57 orientation on e is given by the characteristic map (I^r,^•I^r) → (e,e)^• taking (x1, . . . , xr)to(y1, . . . , yn)whereyi_j =j for 1≤j ≤n−randyp_j =xj for

1≤j≤r.

Proposition 2.5.13.Letri:I^•n→I^•nbe the homeomorphism(x1, . . . , xn)→ (x1, . . . , xi−1,−xi, xi+1, . . . , xn). Then ri is orientation reversing.

Proof. If n = 1, this is obvious. Next, let n = 2 and i = 1, let M_t be the matrix

cos(tπ) sin(tπ) sin(tπ)−cos(tπ)

, and letan:Iⁿ →Bⁿ be the homeomorphism given in Sect. 1.2. Then a⁻₂¹M a2 | I^•2 gives a homotopy from a⁻₂¹f1,−1a2 to r1. For all other cases, embed this matrix in the appropriaten×nmatrix. (For another proof, adapt the first paragraph of the proof of 2.5.2.) Now we discuss the incidence numbers ofIⁿ with its (n−1)-dimensional faces. For =±1, let Fi, be {(x1, . . . , xn) ∈ Iⁿ | xi = }. Each Fi, is an (n−1)-cell ofIⁿ. All cells are canonically oriented.

Proposition 2.5.14.[Iⁿ:F_i,1] =−[Iⁿ:F_i,−1].

Proof. This is clear whenn= 1. Assumen≥2. Recall our permanent choice of homeomorphism,an, for identifyingIⁿwithBⁿ; see Sect. 1.2. Consider the following commutative diagram:

S^{n−1 a}

−1n | //_I^•n

b_n //

r_i

I•ⁿ/(^•Iⁿ−F^◦i,1)oo

∼= Fi,1/F^•i,1

Fi,1

aaCCCCCCCC

Iⁿ⁻¹

h_i,1

ffMMMMM MMMMMMM

h_i,−1

xxqqqqqqqqqqqq

//_In−1/^•Iⁿ⁻¹

k_n−1

∼= //

∼= h_i,−1

∼= h_i,1

OO

Sⁿ⁻¹

Fi,−1

}}||||||||

S^{n−1 a}

−1n | //_I^•n

c_n //_I^•n_/(^•_Iⁿ_−F^◦

i,−1)oo _∼

= Fi,−1/F^•i,−1

Here,hi,:Iⁿ⁻¹→Fi, is the canonical characteristic map

(x1, . . . , xn−1)→(x1, . . . , xi−1, , xi+1, . . . , xn−1) ofFi,, given by 2.5.12, and ri is as in 2.5.13. All other arrows are the obvious quotients or inclusions or are induced by such. By 2.5.13,r_i is orientation reversing.

The required incidence numbers are the degrees off andgwheref [resp.

g] is the composition of maps leading from the top left [resp. bottom left] copy ofSⁿ⁻¹to the right copy ofSⁿ⁻¹. Note thatf =g◦(an◦ri◦a⁻_n¹). By 2.5.13, ri has degree−1. So, by 2.4.19, deg(f) =−deg(g).

Proposition 2.5.15.The mapsb_nandc_nin the above diagram are homotopy equivalences. Hence the incidence numbers in 2.5.14 are±1.

Proof. Leteⁿ(t) ={x∈ Sⁿ | t ≤xn+1 ≤1}. The spaceeⁿ(0) is the upper closed hemisphere, while eⁿ(1) = {north pole}. The reader can easily con- struct a homotopyH :Sⁿ×I →Sⁿ such thatH0 = id, Ht(eⁿ(0)) =eⁿ(t), andH_tmapsSⁿ−eⁿ(0) bijectively ontoSⁿ−eⁿ(t), for allt. The mapH₁ is a quotient map (see Sect. 1.1), and sinceH₁ id,H₁ is a homotopy equiv- alence. Clearly this implies that b_n and c_n are homotopy equivalences. The

second part then follows from 2.4.20.

Proposition 2.5.14 makes a relative statement about incidence numbers.

But even though we know, by 2.5.15, that these incidence numbers are±1, we cannot say which is which (whenn≥2) until we specify the homeomorphism k_n−1 which “was chosen once and for all” earlier in this section. In fact we are only making our choice now:

Convention 2.5.16.For n ≥ 2, the canonical homeomorphism kn−1:Bⁿ⁻¹/Sⁿ⁻²→Sⁿ⁻¹ is to be chosen to make [Iⁿ:Fn,1] = (−1)ⁿ⁺¹. Proposition 2.5.17.[Iⁿ:Fi,1] = (−1)ⁱ⁺¹. Hence⁷ [Iⁿ:Fi,−1] = (−1)ⁱ. Proof. This is clear when n = 1. Assumen ≥2. The orientation onF_n,1 is given byIⁿ⁻¹→F_n,1, (x₁,· · · , x_n−1)→(x₁,· · ·, x_n−1,1); and the orientation on F_n−1,1 is Iⁿ⁻¹ → F_n−1,1, (x₁,· · ·, x_n−1) → (x₁,· · ·, x_n−2,1, x_n−1). Let h : Iⁿ → Iⁿ be the restriction of the linear automorphism of Rⁿ which is the identity in all but the last two coordinates and is given by the matrix [^{0 1}_{1 0}] in those coordinates. Thenhis orientation reversing onI^•n(compare the proof of 2.5.13) but matches up the above orientations on the (n−1)-faces.

By considering the diagram defining incidence numbers, one sees that this implies [Iⁿ:Fn−1,1] =−[Iⁿ, Fn,1]. The conclusion follows, by induction, from

2.5.14, 2.5.15 and 2.5.16.

We now interpret the incidence number [eⁿ_α:eⁿ_β⁻¹] and the mapping degree [eⁿ_α: ˜eⁿ_γ :f] homologically, wheref :X→Y is a cellular map of oriented CW complexes. It is to be understood that singular homology is taken with Z- coefficients. We saw in Sect. 2.3 that the coefficient of ¯hβ∗(λn−1) in∂¯hα∗(λn) is the image of 1 under a certain homomorphismZ →Z. We wish to claim

7 The choice in 2.5.16 which leads to the incidence numbers stated in 2.5.17 is not a universal convention. For example, choices used in [42] lead to [Iⁿ : Fi,1] = (−1)ⁿ⁻ⁱ.

2.5 Orientation and incidence number 59 that this integer is precisely the incidence number [eⁿ_α : eⁿ_β⁻¹]. However, for this to be true some further once-and-for-all choices must be made correctly.

By an easy exercise in singular homology using excision as outlined in Sect.

2.2, there are canonical isomorphisms

H_n^∆(Bⁿ, Sⁿ⁻¹) ^∼= //H_n^∆(Bⁿ/Sⁿ⁻¹, v)oo ^∼= H_n^∆(Bⁿ/Sⁿ⁻¹) ^kn∗ //H_n^∆(Sⁿ) when n ≥ 1. Let κn ∈ H_n^∆(Sⁿ) be the image of the generator λn ∈ H_n^∆(Bⁿ, Sⁿ⁻¹) under this composition of isomorphisms. We also have a ho- momorphism∂_∗:H_n^∆(Bⁿ, Sⁿ⁻¹)→H_n^∆₋₁(Sⁿ⁻¹) which is an isomorphism for n≥2 and a monomorphism whenn= 1; see 2.2.1. Indeed, the image of λ1

under∂_∗ :H₁^∆(B¹, S⁰)→H₀^∆(S⁰)∼=Z⊕Z is the homology class of the sin- gular 0-cycle±({1} − {−1}), with the + or−depending on which generator ofH₁^∆(B¹, S⁰) has been chosen asλ₁.

Now we make our choices. Let κ₀ ∈ H₀^∆(S⁰) be the homology class of {1} − {−1}. Define the generatorsλ_n∈H_n^∆(Bⁿ, Sⁿ⁻¹) andκ_n∈H_n^∆(Sⁿ) by H₀^∆(S⁰)oo oo H₁^∆(B¹, S⁰) ^∼= //H₁^∆(S¹)oo^∼= H₂^∆(B², S¹) ^∼= //H₂^∆(S²)oo · · ·

κ0oo λ1 //κ1oo λ2 //κ2 oo · · · This specifiesλn,n≥1, for the first time. We also defineλ0∈H₀^∆(B⁰) to be the homology class of the singular 0-cycle{0}. With these choices we get:

Proposition 2.5.18.The coefficient of ¯hβ∗(λn−1) in ∂¯hα∗(λn) is the inci- dence number[eⁿ_α : eⁿ_β⁻¹].

Proof. We leave the casen = 1 as an exercise. For n ≥2 the claim can be read off from the following commutative diagram of singular homology groups (∆’s omitted):

H_n(Bⁿ, Sⁿ⁻¹) //

∂∗

Hn(eⁿ_α,^•eⁿ_α) //

∂∗

H_n(Xⁿ, Xⁿ⁻¹)

∂∗

H_n−1(Sⁿ⁻¹) //H_n−1(^•eⁿ_α) //H_n−1(Xⁿ⁻¹, Xⁿ⁻²)

tt oo ^∼=

γ Hn−1(eⁿ⁻¹_γ ,^•eⁿ⁻¹_γ ) projection

uu

Hn−1(Xⁿ⁻¹, Xⁿ⁻¹−^◦eⁿ⁻¹_β )

∼=

oo Hn−1(eⁿ⁻¹_β ,^•eⁿ⁻¹_β )oo

∼=

Hn−1(Bⁿ⁻¹, Sⁿ⁻²)

//Hn−1(Sⁿ⁻¹)

Hn−1(Xⁿ⁻¹/Xⁿ⁻¹−^◦eⁿ⁻¹_β )oo Hn−1(eⁿ⁻¹_β /^•eⁿ⁻¹_β )oo H_n−1(Bⁿ⁻¹/Sⁿ⁻²).

55k

kk kk kk kk k

The point here is that, by our choices,λ_n→κ_n−1on the left andλ_n−1→

κ_n−1 on the right.

Similarly, one proves (exercise):

Proposition 2.5.19.The coefficient of ¯hβ∗(λn)in f#(¯hα∗(λn))is the map-

ping degree[eⁿ_α : ˜eⁿ_β : f].

Exercises

1. LetX andY be oriented CW complexes. GiveX×Y the product orientation.

Prove

(a) [eⁿαטe^mγ :eⁿ_β⁻¹×e˜^mγ] = [eⁿα:eⁿ_β⁻¹] (b) [eⁿ_αטe^m_γ :eⁿ_α×e˜^m_δ⁻¹] = (−1)ⁿ[˜e^m_γ : ˜e^m_δ⁻¹] 2. Prove 2.5.19.

3. Prove then= 1 case of 2.5.18.