Combinatorial fundamental group, Tietze transformations, Van Kampen Theorem

Fundamental Group and Tietze Transformations

3.1 Combinatorial fundamental group, Tietze transformations, Van Kampen Theorem

3

Fundamental Group and Tietze

Certain edge paths can be multiplied. If the final point ofτis also the initial point ofσ theproduct edge path isτ.σ:= (τ1, . . . , τk, σ1, . . . , σm). Whenever a triple product is defined it is obviously associative.

Any edge path has a unique “reduction.” Intuitively this is obtained by removing, step by step, a degenerate edge or an adjacent pairτi,τ_i⁻¹or τ_i⁻¹, τi of edges (unless this leads to the “empty edge path,” in which case the reduction is (v) where v is the initial point of the given edge path). To say this in a correct way, choose an orientation for each 1-cell ofX, and let F be the free group generated by the set of (oriented) 1-cells. Define a function φ from the set of edges of X to F as follows:φ sends all degenerate edges to 1∈F; ifτi is a non-degenerate edge whose orientation agrees with the chosen orientation (so thatτ_iis a generator ofF),φsendsτ_itoτ_iandτ_i⁻¹toτ_i⁻¹(the two meanings ofτ_i⁻¹are clear). Extendφto a functionΦfrom the set of edge paths toF, sendingτ:= (τ₁, . . . , τ_k) toΦ(τ) :=φ(τ₁)φ(τ₂). . . φ(τ_k)∈F. Call τ reduced if eitherΦ(τ) is ak-letter word inF, ork= 1 andτ₁is degenerate.

By convention 1∈ F is a 0-letter word. The reduction of an edge path σ is the unique reduced edge pathτ such thatΦ(σ) =Φ(τ). Clearly, the definition of reduction is independent of the orientations we chose for the 1-cells ofX in definingF.

An edge loop atv is an edge path whose initial and final points arev. A cyclic edge loopis an equivalence class of edge loops under cyclic permutation.

We consider an oriented 2-celle²_γ. In Sect. 2.6 we defined the “homological boundary”∂e²_γ =Σ[e²_γ:e¹_α]e¹_α; we saw in 2.5.8 that the only possible non-zero terms in this sum involve 1-cells contained ine²_γ, and the order in which these 1-cells are considered is irrelevant. Now we define the “homotopical boundary”

∆e²_γ to be the reduced cyclic edge loop obtained by taking those same 1-cells with their incidence numbers, but in strict cyclic order as one goes around the

“edge” of e²_γ in the positive direction. This is made precise in the following paragraphs.

Let h_γ : (B², S¹) → (e²_γ,e^•2_γ) be a characteristic map representing the chosen orientation on e²_γ, and let f = h_γ| : S¹ → X¹ be the correspond- ing attaching map. First, suppose f⁻¹(X⁰) = ∅. Each path component of S¹−f⁻¹(X⁰) is an open interval mapped byf into some^◦e¹_β. By elementary analysis there are countably many such open intervals: label themI^◦j, j≥1.

Lete¹_β

j be the unique 1-cell such thatf(I^◦j)⊂^◦e¹_β

j. Proposition 3.1.1.f maps ^◦Ij onto^◦e¹_β

j for at most finitely manyj.

Proof. Let the 1-cells of X be {e¹_α | α ∈ A}. Pick zα ∈ ^◦e¹_α. There is an open cover of X¹ consisting of X¹− {zα | α∈ A} =: U and each e^◦1_α. The open cover{f⁻¹(U)} ∪ {f⁻¹(^◦e¹_α)| α∈ A} of S¹ has a finite subcoverU = {f⁻¹(U), f⁻¹(^◦e¹_α

1), . . . , f⁻¹(^◦e¹_α

n)}. If the proposition were false, at least one

3.1 Fundamental group, Tietze transformations, Van Kampen Theorem 75

of the setsf⁻¹(^◦e¹_α

i) could be written as ∞ k=1

I◦j_k with alljk’s distinct and no I◦_jk covered by any member ofU other thanf⁻¹(e^◦1_α

i); this is because eachI^◦_jk meetsf⁻¹(zα_i). Replacing f⁻¹(e^◦1_α

i), inU, by all the intervalsI^◦j_k, we would have an open cover ofS¹ having no finite subcover, a contradiction sinceS¹

is compact.

For convenience, label the intervals I^◦j so that I^◦1, . . . ,I^◦k are those men- tioned in 3.1.1 (mapped onto 1-cells). Arrange further that I^◦₁, . . . ,^◦I_k are labeled in cyclic order with respect to positive rotation onS¹. First, assume there is at least one such interval (i.e.,k≥1). For each 1≤j ≤k, we define alocal incidence numberij=±1of e²_γ with e¹_β

j along ^◦Ij as the degree of the self-map ofS¹ indicated in the following commutative diagram:

clI^◦j f|

−−−−→ e¹_β

⏐ j

⏐^pj ⏐⏐^qj

S¹ −−−−→^rj e¹_β

j/e^•1_β

j

h

←−−−−_∼βj

= B¹/S⁰ −−−−→^k_∼¹

= S¹.

Here,p_jis the restriction to cl^◦I_j of a degree 1 mapS¹→S¹which is constant onS¹−I^◦_jand is injective on^◦I_j;q_jis the quotient map;r_j is the unique map making the left square commute; h_β

j and k₁ are as in Sect. 2.5. Clearly, f| is homotopic, rel frI^◦j, to a map which takesI^◦j homeomorphically ontoe^◦1_β

j, hencerj is homotopic to a homeomorphism. By 2.4.20,ij =±1, as claimed.

The homotopical boundary of the oriented 2-cell e²_γ is the reduced cyclic edge loop ∆e²_γ represented by the reduction of the edge loop (τ_βⁱ¹

1, . . . , τ_β^ik

k), whereτβ_j is the edgee¹_β

j with the preferred orientation.

Proposition 3.1.2.

τ_βⁱ¹

1, . . . , τ_β^ik

k

is indeed an edge loop, and is unique up to cyclic permutation.

Proof. Consider¹I^◦jwherej > k. Sincef(^◦Ij) is a proper subset of^◦e¹_β

j,f(frIj) is a single point of^•e¹_β

j. Thus there is a mapg:S¹→X¹, agreeing withf on I◦j when j≤k, such thatg

⎛

⎝S¹−

⎛

⎝^k

j=1

I◦j

⎞

⎠

⎞

⎠⊂X⁰. Indeed, adjustingg on

1 It is not to be thought that the labels> k occur in cyclic order: there might be infinitely many intervalsI^◦j, perhaps infinitely many in each quadrant ofS¹.

each^◦Ij (1≤j ≤k) by a homotopy rel fr I^◦j, we may assume thatg mapsI^◦j

homeomorphically ontoe^◦_βj when 1≤j ≤k. It is then obvious that we have

a cyclic edge loop as claimed.

There remain two cases. If f⁻¹(X⁰) = ∅, then there is a single 1-celle¹_β such that f(S¹)⊂ ^◦e¹_β, and we define ∆e²_γ to be the degenerate cyclic edge loop represented solely by the initial point ofe¹_β. Iff⁻¹(X⁰)=∅ but k= 0, there is a single vertexvcommon to all the 1-cells which meetf(S¹), and we define∆e²_γ to be the degenerate cyclic edge loop (v).

Clearly, a change of orientation on the celle²_γ causes an inversion of ∆e²_γ. That∆e²_γdepends only on the orientation rather than a specific characteristic maphγ is an exercise.

We can now define equivalence of edge paths. Let σandτ be edge paths in the oriented CW complex X having the same initial point and the same final point. Write σ τ iff either τ is the reduction of σ, or there is an (oriented) 2-celle²_γ such that σ.τ⁻¹ = λ.µ1.ν.ν⁻¹.µ2.λ⁻¹ where µ1.µ2 is an edge loop representing the cyclic edge loop ∆e²_γ; see Fig. 3.1. We will call either of these an elementary equivalence between σ and τ. The relation generates an equivalence relation on the set of edge paths inX, which we call equivalence.

m

2 1

e2_g

n l

s

t m

Fig. 3.1.

The next three propositions are left as exercises:

Proposition 3.1.3.Products of equivalent edge paths are equivalent. Inverses

of equivalent edge paths are equivalent.

Proposition 3.1.4.Let the edge path τ have initial point v1 and final point v2. Then τ is equivalent to both(v1).τ andτ.(v2).

3.1 Fundamental group, Tietze transformations, Van Kampen Theorem 77 Proposition 3.1.5.If τ is an edge path with initial point v1 and final point v2, thenτ.τ⁻¹ is equivalent to (v1) andτ⁻¹.τ is equivalent to (v2).

Letv be a vertex ofX. Our discussion shows that the set of equivalence classes of edge loops atv is a group whose multiplication is induced by the product operation, whose identity is the equivalence class of v, and whose inversion is induced by the inverse operation. This group is thefundamental group of X based at v. It is denoted byπ₁(X, v).

Proposition 3.1.6.The definition of equivalence of edge paths is independent of the orientation chosen for X. In particular, π1(X, v) does not depend on

the orientation ofX.

Proposition 3.1.6 clarifies the role of the chosen orientation of X in this section. We must choose an orientation in order to define equivalence of edge loops at v, but the definition of equivalence turns out to be independent of this choice.

Proposition 3.1.7.π1(X, v)depends only on the2-skeleton of the path com-

ponent of X containing v.

Theorem 3.1.8.Let X be an oriented CW complex having only one vertex, v. LetF be the free group generated by the set W of (oriented) 1-cells ofX, letRbe the set of (oriented)2-cells ofX, for each e²_γ ∈Rletτ:=τ(γ)be an edge loop representing∆e²_γ, and letρ:R→F be the function² e²_γ→Φ(τ(γ)).

Then W |R, ρis a presentation of π1(X, v).

Proof. LetG=W | R, ρ. It is enough to define an epimorphismΨ : G→ π₁(X, v) and a functionΦ:π₁(X, v)→Gsuch thatΦ◦Ψ = id_G.

Each edge of X is an edge-loop at v. Define ψ : W → π₁(X, v) to take σ∈W to the equivalence class of the edge-loop (σ);ψextends uniquely to an epimorphismΨ :F →π1(X, v). Fix e²_γ ∈R. Let τ :=τ(γ) = (τ₁ⁱ¹, . . . , τ_nⁱⁿ), where eachτj ∈W andij =±1. Using the notation introduced earlier in this section,Φ(τ) =φ(τ1)ⁱ¹. . . φ(τn)ⁱⁿ∈F. If the reduced form of this element of Fisσ^j₁¹. . . σ_k^jk(eachσm∈W, eachjm=±1), thenΨ(Φ(τ)) is the equivalence class of the edge loop (σ^j₁¹, . . . , σ^j_k^k). This in turn is the reduction of the edge loop (τ₁ⁱ¹, . . . , τ_nⁱⁿ), which is equivalent to the edge loop (v). So Ψ(Φ(τ)) = 1∈π1(X, v). Thus there is an induced epimorphismΨ :G→π1(X, v).

To define Φ, consider Φ:{edge loops in X at v} →F; Φmaps an edge loop and its reduction to the same element of F. If σ and η are two edge loops such thatσ.η⁻¹=λ.µ₁.ν.ν⁻¹.µ₂.λ⁻¹, as above, whereµ₁.µ₂ represents

∆e²_γ, then Φ(σ)Φ(η)⁻¹ = Φ(λ)Φ(µ₁.µ₂)Φ(λ)⁻¹ which is conjugate to Φ(τ), sinceτ and µ₁.µ₂ differ by a cyclic permutation. Thus Φinduces a function Φ:π₁(X, v)→Gwhich is clearly a left inverse for Ψ.

2 RecallΦ:{edge paths} →F, above.

Thus, Example 1.2.17 gives a procedure for constructing a 2-dimensional CW complex having just one vertex, and any prescribed group as fundamental group.

Corollary 3.1.9.Let X be an oriented 1-dimensional CW complex having only one vertexv. Then the (oriented)1-cells are edge loops whose equivalence

classes freely generateπ₁(X, v).

Corollary 3.1.10.Let X be a CW complex having one vertex v and no 1-

cells. Thenπ1(X, v) ={1}.

IfAis a subcomplex ofX, withv∈A, the inclusion mapi: (A, v)→(X, v) induces an obvious homomorphism i_# : π₁(A, v) → π₁(X, v). The example X = B², with cells {v}, S¹ and B², and A = S¹, shows that ker i_# need not be trivial (by 3.1.8 and 3.1.9). Indeed, this homomorphismi#is a special case of “the homomorphism induced by a map,” but that definition is best left until Sect. 3.3.

There is a sense in which π1(X, v) is independent of the base vertex v, provided that X is path connected. Letτ be an edge path inX with initial point v and final point v; τ induces a function hτ : π1(X, v) → π1(X, v), [σ] → [τ⁻¹.σ.τ], where [σ] ∈ π1(X, v) denotes the equivalence class of the edge loopσatv. Clearly, we have

Proposition 3.1.11.h_τ is well defined and is a group isomorphism whose

inverse ish_τ−1.

The CW complex X is simply connected if X is path connected and π1(X, v) is trivial for some (equivalently, any) vertexv ofX.

A tree is a non-empty CW complex,T, of dimension≤1 in which, given verticesv₁andv₂ofT, there is exactly one reduced edge path inT with initial vertexv₁ and final vertexv₂.

Proposition 3.1.12.Every tree is simply connected. Every simply connected CW complex of dimension ≤1 is a tree. Every tree is contractible.

Proof. A tree T is clearly simply connected, for if (τ₁, . . . , τ_n) is a reduced edge loop in T at the vertex v then n = 1. Conversely, let X be simply connected and of dimension≤1 and let v be a vertex. In the absence of 2- cells, the equivalence relation on edge loops atvdefiningπ1(X, v) boils down to reduction: no two distinct reduced edge loops are equivalent. So there is only one reduced edge loop. It follows that there is only one reduced edge path fromv to each vertex ofX.

The fact that trees are contractible follows easily from 1.3.14.

We say that T is a tree in X if T is a tree and is a subcomplex of X; T is amaximal tree if there is no larger subcomplex of X which is a tree. If X is path connected it follows that a maximal tree in X contains X⁰ and, conversely, any tree inX containingX⁰is maximal.

3.1 Fundamental group, Tietze transformations, Van Kampen Theorem 79 Proposition 3.1.13.Let(X, A)be a CW pair withX andApath connected.

Let TA be a maximal tree inA. There is a maximal tree, TX, in X such that TX∩A=TA.

Proof. LetT denote the set of treesT in X such thatT ∩A=TA. This set T is non-empty and is partially ordered by inclusion. Let {Ti} be a linearly ordered subset ofT and let ˜T =

i

Ti. Clearly, ˜T ∈ T. By Zorn’s Lemma, T

contains a maximal element,TX, as required.

Next, we review the rules for altering a presentation of a group. LetW | R, ρbe a presentation of the group³ G. A Tietze transformation of Type I replacesW |R, ρbyW |R, ρwhere:Sis a set,ψ:S→N(ρ(R))⊂F(W) is a function,R=R

S, andρ :R →F(W) agrees withρonRand withψ onS. ATietze transformation of Type II replacesW |R, ρbyW |R, ρ, where: S is a set, W = W

S, ψ : S → F(W) ⊂ F(W) is a function, R =R

S, andρ :R →F(W) agrees with (inclusion)◦ρonR, and⁴with s→s.ψ(s)⁻¹ onS.

Example 3.1.14.Starting with a | a³, a presentation of the cyclic group of order 3, a Type I transformation might change this to a | a³, a⁶; a Type II transformation might change this latter presentation to a, b, c,| a³, a⁶, ba, ca².

The following is well-known (see [107] or [106] for a proof):

Theorem 3.1.15.Tietze transformations of either type applied to a presen- tation ofGyield another presentation of G. Conversely, ifW1|R1, ρ1and W2|R2, ρ2 are presentations ofG, then there are presentations and Tietze transformations as follows:

W1|R1, ρ1^{Type II}+3_{W|R, ρ} ^{Type I}+3_{W|R, ρ}ks^{Type I} _{W|R, ρ}ks^{Type II}_{W2|R2, ρ2.} Moreover, if the given presentations are finite, then the intervening presenta-

tions can be chosen to be finite.

We are now ready to generalize 3.1.8 by explaining how to read off a pre- sentation of the fundamental group of any path connected CW complex from the 2-skeleton. Note that (by 3.1.13 applied withAa one-vertex subcomplex) such complexes always contain maximal trees.

Theorem 3.1.16.Let X be an oriented path connected CW complex, let T be a maximal tree in X and let v be a vertex of X. Let F be the free group

3 Recall the notation for presentations introduced in 1.2.17.

4 Note that by our conventionsS⊂F(S)⊂F(W).

generated by the setW of (oriented)1-cells ofX. LetRbe the set of (oriented) 2-cells ofX and letS be the set of (oriented)1-cells ofT. For eache²_γ ∈R, let τ :=τ(γ) be an edge loop in X representing ∆(e²_γ). Let ρ:R

S →F take each e²_γ ∈R toΦ(τ(γ)), and take each σ∈S to the one-letter word σ∈F. Then W |R

S, ρis a presentation ofπ1(X, v).

Proof. We claimπ₁(X, v)∼=π₁(X/T ,¯v) ∼=W −S |R,ρ¯ ∼=W |R S, ρ, where ¯vis the only vertex ofX/T,q:F(W)→F(W−S) is the epimorphism of free groups sendingwtowwhenw∈W−S, andwto 1 whenw∈S, and

¯

ρ=q◦ρ|:R→F(W −S).

There is an obvious epimorphism α: π1(X, v)→ π1(X/T ,¯v). The proof that α is an isomorphism boils down to showing that if ∆e²_γ is represented by the edge loop (τ₁ⁱ¹, . . . , τ_nⁱⁿ) in X, then the edge loop inX/T obtained by deleting those edgesτj which lie inT represents∆¯e²_γ, where ¯e²_γ is the 2-cell of X/T corresponding toe²_γ, which is clear. This is our first isomorphism. The second isomorphism comes from 3.1.8. The third isomorphism comes from Tietze transformations

W−S|R,ρ¯ ^{Type II}+3_W|R^‘_{S, ρ} ^{Type I}+3_{W |}(R‘ S)‘

R, ρks^{Type I} _{W |S}^‘_{R, ρ} whereρ = ¯ρonR,ρ= id onS,ρ=ρ onR

S, andρ=ρon the second

copy ofR.

Corollary 3.1.17.Let X be a path connected CW complex. If X¹ is finite, then π1(X, v) is finitely generated. If X² is finite, then π1(X, v) is finitely

presented.

Another corollary of 3.1.16 expresses the fundamental group of a path connected CW complex in terms of the fundamental groups of path connected subcomplexes:

Theorem 3.1.18. (Seifert-Van Kampen Theorem)⁵Let the CW complex X have subcomplexes X1, X2 and X0 such that X = X1 ∪X2 and X0 = X1∩X2. Assume X1,X2 andX0 are path connected. Letv∈X0 and let i1: X0 →X1,i2 :X0→X2,j1 :X1→X, andj2:X2 →X be the inclusions.

Then⁶ (j1#, j2#) :π1(X1, v)∗π1(X2, v)→π1(X, v)is an epimorphism whose kernel is the normal closure of{i1#(g)i2#(g)⁻¹|g∈π1(X0, v)}.

Proof. By 3.1.13, there is a maximal treeT_X such that T_Xi :=X_i∩T_X is a maximal tree inX_ifori= 0,1,2. LetW_X |R_X

S_X, ρ_Xbe a presentation of π₁(X, v) as in 3.1.16. Using restrictions of the items in this presentation,

5 Theorem 6.2.11 is a useful generalization of this.

6 Notation:G∗H denotes the free product ofGandH.

3.1 Fundamental group, Tietze transformations, Van Kampen Theorem 81 form presentationsWX_i |RX_i

SX_i, ρX_iforπ1(Xi, v),i= 0,1,2. Abusing notation, we have obvious homomorphisms

WX₀ |RX₀

SX₀, ρX₀ −−−−→ ^i1# WX₁ |RX₁

SX₁, ρX₁

i_2#

⏐⏐

⏐⏐^j1#

WX₂ |RX₂

SX₂, ρX₂ −−−−→

j_2# WX|RX

SX, ρX.

Note that W_X0 ⊂ W_Xi ⊂ W_X, R_X0 ⊂ R_Xi ⊂ R_X, S_X0 ⊂ S_Xi ⊂ S_X, for i= 1,2. FOR THIS PROOF ONLY we introduce a convention. Let ˜W_X0, ˜R_X0 and ˜S_X0 be copies of the setsW_X0,R_X0 andS_X0 respectively. Ifσ∈W_X0 we denote the corresponding element of ˜W_X0 by ˜σ, etc.

The desired result is obtained by applying two Tietze transformations:

W_X |R_X

S_X, ρ_X^{Type II}+3WX

W˜X₀ |RX

SX

W˜X₀, ρ^{Type I}+3 WX

W˜X₀ |RX

SX

W˜X₀

( ˜RX₀

S˜X₀), ρ where ρ = (inclusion)◦ρ_X on R_X

S_X, and, for each ˜σ ∈ W˜_X0, ρ(˜σ) =

˜

σ.σ⁻¹; ρ =ρ on R_X

S_XW˜_X0, and, for each ˜τ ∈ R˜_X0S˜_X0, ρ(˜τ) = ρ_X(τ)∈F(W_X)⊂F(W_XW˜_X0).

The latter presentation can be rewritten as WX₁

(WX−WX₁)W˜X₀ |RX₁

(RX−RX₁) SX₁

(SX−SX₁)W˜X₀

( ˜RX₀

S˜X₀), ρ.

If we identify (WX−WX₁)W˜X₀ =WX₂, (RX−RX₁)R˜X₀ =RX₂, (SX− S_X1)S˜_X0 = S_X2, and W_X0 = the appropriate subset of W_X1, then this becomes:

WX₁

WX₂ |RX₁

SX₁

RX₂

SX₂

W˜X₀,ρ¯ where ¯ρ = (inclusion)◦ρ_Xi on R_Xi

S_Xi for i = 1 and 2, and for each

˜

σ∈W˜_X0, ¯ρ(˜σ) = ˜σ.σ⁻¹. This is clearly a presentation of WX₁|RX₁

SX₁, ρX₁ ∗ WX₂ |RX₂

SX₂, ρX₂/N

1#(g)i2#(g)⁻¹|g∈WX₀|RX₀

SX₀, ρX₀}.

Some notation is useful here. IfG1 andG2 are groups and ifS is a subset of the free productG1∗G2, then we write G1, G2|Sfor a presentation of G1∗G2/N(S). The notation indicates that generating sets forG1andG2are on the left of the vertical bar, and relation data forG1andG2are on its right, together with the additional relations S. In this notation, the conclusion of the Seifert-Van Kampen Theorem 3.1.18 can be rewritten⁷:

7 In the Appendix to this section we discuss this and other variants of “presenta- tion”.

whereNis the normal closure of{i

π1(X, v)∼=

π1(X1, v), π1(X2, v)|i1#(g)i2#(g)⁻¹, ∀g∈π1(X0, v) In the special case wherei_1#and i_2# are monomorphisms this becomes the free product with amalgamation which is defined in Sect. 6.2.

To end this section, we point out a variation on 2.7.4 showing the relation- ship betweenπ1(X, v) andH1(X;Z). Let τ := (τ₁ⁱ¹, . . . , τ_k^ik) be an edge loop inX atvwhereτj has the preferred orientation (X is oriented) andij=±1.

Then ˜h(τ) :=

k j=1

i_jτ_j is a 1-cycle inX.

Theorem 3.1.19.This function ˜hinduces a homomorphism h:π₁(X, v)→ H₁(X;Z)whose kernel is the commutator subgroup of π₁(X, v). IfX is path connected,his an epimorphism.

Proof. Clearly,his a well defined homomorphism. We may assume thatX is path connected: otherwise we could work with the path component containing v. We first deal with the special case in whichXhas only one vertex. Then the result follows from 3.1.8, since the effect of abelianizing the groupW |R, ρ in that theorem is to produce the abelian⁸groupH₁(X;Z).

In the general case, pick a maximal tree,T, inX. Consider the diagram:

π₁(X, v) −−−−→_∼^α

= π₁(X/T ,v)¯

⏐⏐

^h ⏐⏐^h H₁(X;Z) −−−−→^q∗ H₁(X/T;Z)

Here,αis the isomorphism discussed in the proof of 3.1.16,h is the version ofhforX/T, andq_∗is the homomorphism induced by the (cellular) quotient mapq:X →X/T. The diagram clearly commutes. By the special case,h is an epimorphism whose kernel is the commutator subgroup. It only remains to show thatq_∗ is an isomorphism. The discussion of relative homology in Sect.

2.8 and Sect. 2.9 shows that the following diagram commutes and the top line is exact:

H1(T;Z) //H1(X;Z) ^j# //

q_∗

&&

NN NN NN NN NN

N H1(X, T;Z) //H˜0(T;Z)

H1(X/T;Z)

∼

=

OO

By 3.1.12,j#is an isomorphism, hence alsoq_∗. Thus, for path connected X,H₁(X;Z) is theabelianization ofπ₁(X,∗).

The homological, or abelianized, version of the Seifert-van Kampen theo- rem is:

8 The free abelian group generated by W is Z₁(X;Z), and the abelianization of

∆e²_γ is∂e²_γ, so the result isZ₁(X;Z)/B₁(X;Z).

3.1 Fundamental group, Tietze transformations, Van Kampen Theorem 83 Proposition 3.1.20.With hypotheses as in 3.1.18, j1∗+j2∗ : H1(X1;Z)⊕ H1(X2;Z)→H1(X;Z)is an epimorphism whose kernel is {i1∗(z)−i2∗(z)| z∈H1(X0;Z)}.

Proof (First Proof ).Apply 3.1.18 and 3.1.19.

Proof (Second Proof ).The Mayer-Vietoris sequence (Sect. 2.8) gives an exact sequence

H1(X0;Z) −−−−−−→^{(i1∗,−i2∗)} H1(X1;Z)⊕H1(X2;Z) −−−−−→^j1∗+j2∗ H1(X;Z) −−−−→^∂∗ H0(X0;Z) −−−−−−→^{(i1∗,−i2∗)} H0(X1;Z)⊕H0(X2;Z) The rightmost arrow is a monomorphism, by 2.7.2 and the definition of i_1∗.

Hence∂_∗= 0. The result follows by exactness.