Elementary Geometric Topology
5.1 Review of topological manifolds
5
126 5 Elementary Geometric Topology
Much easier than these results is the following proposition which is essen- tially Exercise 3 in Sect. 2.2:
Proposition 5.1.6.(a)Letm=n. No non-empty open subset ofRnis home- omorphic to an open subset ofRm.
(b) Letx∈Rn−1⊂Rn+. No neighborhood ofxin Rn+ is homeomorphic to
Rn.
LetM be a manifold of dimensionn. TheboundaryofM is{x∈M |some neighborhood ofx is homeomorphic to Rn+}; this subset is denoted by ∂M. The setM−∂M, abbreviated toM◦, is theinterior ofM; it consists of all the points ofM having neighborhoods homeomorphic toRn. The manifoldM is closed ifM is compact and∂M is empty.M isopen if each path component ofM is non-compact and∂M is empty.1
Proposition 5.1.6 has a number of consequences. IfM is non-empty thenM◦ is non-empty. Hence, by considering a neighborhood of a point ofM◦ , one sees that the dimension ofM is well defined. Moreover,∂M∩M◦ =∅. A manifold of dimensionnis called ann-manifold; a manifold of dimension 2 is also called a surface. Note that since∂M is topologically characterized, a homeomorphism between manifolds carries boundary to boundary and interior to interior. If M carries the structure of a CW complex, the dimension (in the sense of Sect.
1.2) of that CW complex isn.
Here are some easily deduced properties of an n-manifoldM: ∂M is an (n−1)-manifold; ∂(∂M) = ∅; ∂M is a closed subset of M; ∂M is nowhere dense inM;M◦ is open inM;M◦ is dense inM. Manifolds are locally compact.
Remark 5.1.7.Consider the special cases of 5.1.3 and 5.1.4 in whichn= 3. If A is an arc (i.e., is homeomorphic toB1) and S is homeomorphic to S2, we haveH1∆(S3−A) = 0 =H1∆(S3−S). However, there is an arcA⊂S3(the Fox-Artin Arc) whose complement is not simply connected; and there is a 2-sphereS⊂S3(the Alexander Horned Sphere) one of whose complementary components is not simply connected. For more on these, see, for example, [138].
Let M be a path connectedn-manifold and let B ⊂M◦ be ann-ball. By 5.1.1, intB is an open subset ofM, but it does not follow thatM −intB is a manifold.2 An n-ballB inM is unknotted ifM−intB is a manifold. IfB
1 The reuse of the terms “interior”, “closed” and “open” is unfortunate but en- trenched. More troublesome is the reuse of “boundary”; in addition to the two uses of that word in this book (here and in Sect. 2.1) many authors also use it for what we have called the “frontier”. As we remarked in Sect. 1.2, the word
“boundary” is also used for the subset•eof a cellein a CW complex.
2 For exampleB might be the closure of the component of (S3– Alexander Horned Sphere) which is simply connected – see 5.1.7.
andCare unknottedn-balls inM there is a homeomorphism3ofM takingB ontoC; this is a consequence of the Generalized Schoenflies Theorem (see [45]).
Unknotted balls are needed for the “connected sum construction”: letM1and M2 be path connected n-manifolds, letB1 and B2 be unknottedn-balls4 in M◦1andM◦2 respectively with boundariesS1andS2, and leth:S1→S2be a homeomorphism between these (n−1)-spheres. The relationx∼h(x) for all x∈S1 defines an equivalence relation on cl(M1−intB1)
cl(M2−intB2) and the resulting quotient space is called the connected sum ofM1 and M2, denoted M1#M2. It is a manifold because the balls are unknotted. Up to homeomorphism this is independent of the choices ofB1,B2 andh. See Fig.
5.1.
M M
S S
M # M
1 2
1
1 2
2 h
Fig. 5.1.
Example 5.1.8.Theclosed orientable surfaceTgis defined forg∈Nby:T0= S2, T1 = T2 (the 2-torus), Tg+1 = Tg#T1. This number g is the genus of the surface. This surfaceTg is also called asphere with ghandles. Theclosed
3 Indeed,B can be carried ontoCby an ambient isotopy, i.e., a homotopy through homeomorphisms starting at idM.
4 It is easy to find unknottedn-balls in Rn, hence inM◦ for anyM.
128 5 Elementary Geometric Topology
non-orientable surface Uh is defined forh≥1 by:U1 =RP2 (the projective plane),Uh+1=Uh#U1. This surfaceUhis called5asphere withhcrosscaps. It is well-known, see for example [110] or [151], that every closed path connected surface is homeomorphic to some Tg or some Uh. These surfaces are CW complexes as follows. LetK(g) be the presentation complex of
a1, b1,· · ·, ag, bg|[a1, b1][a2, b2]· · ·[ag, bg]
where [a, b] denotes the commutatoraba−1b−1, and letL(h) be the presenta- tion complex of
a1,· · · , ah|a21a22· · ·a2h.
The underlying spaces of the CW complexesK(g) andL(h) areTg and Uh
respectively. In particular we see presentations of thesurface groups, namely the fundamental groups ofTg and Uh. The abelianizations are pairwise non- isomorphic, so no two of the surfaces Tg and Uh have the same homotopy type. All these surfaces except T0 = S2 and U1 = RP2 are aspherical (see Exercise 7) so their fundamental groups have typeF.
Example 5.1.9.IfM is a compact surface with non-empty boundary then each path component of∂Mis a circle, and ifCis one of those circles then the space obtained fromM by attaching a 2-cell using as attaching map an embedding S1 → M whose image is C is again a surface. Thus, by attaching finitely many 2-cells in this wayM becomes a closed surface, i.e., becomes aTg or a Uh. LetTg,d andUh,d denote the surfaces obtained by removing the interiors of d unknotted6 2-balls. Then every compact path connected surface whose boundary consists ofdcircles is homeomorphic toTg,dor toUh,d. Whend >0, Tg,d and Uh,d contain graphs as strong deformation retracts; hence they are aspherical with free fundamental groups, by 3.1.9 and 3.1.16.
Exercises
1. Prove 5.1.3–5.1.5. Then prove 5.1.1.
2. Show that whenp: ¯X →X is a covering projection, ¯X is a manifold iffX is a manifold.
3. Show that every path connected 1-manifold is homeomorphic to S1, I, R or [0,∞).
4. Show that ifM is ann-manifold thenM#Snis homeomorphic toM. 5. Show thatTg+1,d is homeomorphic toTr,k#Tg+1−r,d−k+2.
6. For n-manifolds M1 andM2, show thatπ1(M1#M2) ∼=π1(M1)∗π1(M2) when n >2. Discuss the casesn≤2.
7. Forg≥3 show thatTg is a (g−1) to 1 covering space ofT2.
5 One expressesTg+1=Tg#T1 [resp. Uh+1=Uh#U1] by sayingTg+1is obtained from Tg byattaching a handle [resp. Uh+1 is obtained fromUh byattaching a crosscap].
6 By the Schoenfliesz Theorem (see [138]) every 2-ball in a surface is unknotted.
8. Show thatTg is a 2 to 1 covering space ofUg+1.
9. By considering the universal cover of K(2) (see Example 5.1.8) show that the universal cover ofT2 is homeomorphic toR2. Deduce that the universal covers of all the path connected closed surfaces exceptS2 andRP2 are homeomorphic toR2. (In the terminology of Ch. 7, all such surfaces are aspherical.)
10. LetGact freely and cocompactly on a path connected orientable open surfaceS.
Prove thatH1(S;Z) is finitely generated as aZG-module ifGis finitely presented, and it is not finitely generated as aZG-module ifGis finitely generated but does not have typeF P2 overZ.