Basic Examples
Definition (R n ) We define real or Euclideann-space, denoted by R n , as the set
We begin by looking at some basic subspaces ofR n
Definition (standard n-disk) The n-dimensional disk, denoted D n is de- fined as
For example, D 1 = [0,1] D 1 is also called the unit interval, sometimes denoted byI.
Definition (standard n-ball, standard n-cell) The n-dimensional ball or cell, denoted B n , is defined as:
Fact 1.1 The standard n-ball and the standard n-disk are compact and homeomorphic.
Definition(standard n-sphere) The n-dimensional sphere, denoted S n , is defined as
As usual, the term n-sphere will apply to any space homeomorphic to the standardn-sphere.
Question 1.2 Describe S 0 , S 1 , and S 2 Are they homeomorphic? If not,are there any properties that would help you distinguish between them?
Simplices
In this article, we will explore manifolds, specifically k-manifolds, which consist of segments that locally resemble R^k and are assembled in a coherent manner Our focus will be on manifolds constructed from triangles and their higher-dimensional counterparts as fundamental building blocks.
Since k-dimensional “triangles” in R n (called simplices) are the basic building blocks we will be using, we begin by giving a vector description of them.
A 1-simplex, also known as an edge, is defined by two points, v0 and v1, in R^n When treated as vectors originating from a common point, the set σ1 = {αv1 + (1−α)v0 | 0 ≤ α ≤ 1} represents the straight line segment connecting v0 and v1 This segment can be denoted as {v0 v1} or {v1 v0}, with the order of the vertices being irrelevant The vertices v0 and v1 are referred to as 0-simplices.
Simplicial Complexes
Definition (2-simplex) Let v0, v1, and v2 be three non-collinear points in
The set σ² represents a 2-simplex, specifically a triangle formed by the vertices v₀, v₁, and v₂ It is defined by the linear combination σ² = {λ₀v₀ + λ₁v₁ + λ₂v₂ | λ₀ + λ₁ + λ₂ = 1 and 0 ≤ λᵢ ≤ 1 for all i = 0, 1, 2} The edges of this triangle are {v₀v₁}, {v₁v₂}, and {v₀v₂} The notation {v₀v₁v₂} signifies the same 2-simplex σ², highlighting that the order of the vertices does not affect its representation.
Note that the plural of simplex issimplices.
Definition (n-simplex and face of a simplex) Let {v 0 , v2, , vn} be a set affine independent points in R N Then an n-simplex σ n (of dimension n), denoted {v 0 v 1 v 2 v n }, is defined to be the following subset ofR N : σ n ( λ 0 v 0 +λ 1 v 1 + +λ n v n n
An i-simplex, defined by any subset of i + 1 vertices from the vertices of σn, represents an i-dimensional face of σn When the vertex vm is removed from the list of vertices of σn, the resulting face is typically denoted as {v0, v1, v2, , vc m, , vn}, which is classified as an (n−1)-simplex.
Exercise 1.3 Show that the faces of a simplex are indeed simplices.
Fact 1.4 The standard n-ball, standard n-disk and the standard n-simplex are compact and homeomorphic.
We will use the terms n-disk, n-cell, n-ball interchangeably to refer to any topological space homeomorphic to the standard n-ball.
Simplices can be assembled to create polyhedral subsets of R n known as complexes These simplicial complexes are the principal objects of study for this course.
A finite simplicial complex is defined as a finite collection of simplices in R^n, denoted as T, where each simplex's faces are also included in T, and any two simplices are either disjoint or share a face The subset K of R^n, formed by the union of all simplices in T, is referred to as the finite simplicial complex with triangulation T, represented as (K, T) The set K is known as the underlying space of the simplicial complex, and if n represents the maximum dimension of the simplices in T, then the complex (K, T) is classified as having dimension n.
Example 1.Consider(K, T)to be the simplicial complex in the plane where
SoK is a filled in triangle and a hollow triangle as pictured.
Exercise 1.5 Draw a space made of triangles that is nota simplicial com- plex, and explain why it is not a simplicial complex.
We began by constructing spaces using simplices as foundational elements However, the question arises: how can we decompose a given space into simplices? If J is a topological space that is homeomorphic to K, where K serves as the underlying space of a simplicial complex (K, T) in R^m, we refer to J as triangulable For Exercise 1.6, demonstrate that the specified space is triangulable by providing a suitable triangulation.
Definition (subdivision) Let (K, T) be a finite simplicial complex Then
T 0 is a subdivision of T if (K, T 0 ) is a finite simplicial complex, and each simplex in T 0 is a subset of a simplex in T.
Example 2 The following picture illustrates a finite simplicial complex and a subdivision of it.
There is a standard subdivision of a triangulation that later will be useful:
1 Let σ 2 be a 2-simplex with vertices v 0 , v 1 , and v 2 Then p = 1 3 v 0 +
2 Let T be a triangulation of a simplicial 2-complex with 2-simplices {σ i } k i=1 The first derived subdivision of T, denoted T 0 , is the union of all vertices of T with the collection of 2-simplices obtained from
The process involves dividing each σ i in the triangulation T into six segments, which includes the respective edges and vertices Additionally, edges that do not form a face of a 2-simplex are split into two edges, resulting in new vertices that represent the barycenter of each σ i and the center of each edge in T This leads to a second derived subdivision, denoted as T 00, which is the first derived subdivision of T 0.
Example 3 Figure 1.2 illustrates a finite simplicial complex and the second derived subdivision of it.
The concept of the real line and the Euclidean spaces produced from the real line are fundamental to a large part of mathematics So it is natural to
Figure 1.1: Barycentric subdivision of a 2-simplex
In topology, researchers focus on spaces that exhibit characteristics similar to Euclidean spaces, particularly those that are locally like R² A prime example is the 2-sphere, which resembles the Earth's surface, especially in flat regions such as Kansas or the vastness of the Pacific Ocean When on a ship in the middle of the ocean, the horizon appears flat, akin to living on a plane, which exemplifies Euclidean 2-space The term for a space that is locally homeomorphic to R² is known as a surface or 2-manifold Both the 2-sphere and the torus, resembling a doughnut or inner-tube, fall under this category of surfaces.
Definition(2-manifold or surface) A2-manifold or surfaceis a separable,metric space Σ 2 such that for each p ∈Σ 2 , there is a neighborhood U of p that is homeomorphic toR 2
For now, we will restrict ourselves to 2-manifolds that are subspaces of R n and that are triangulated.
Definition (triangulated 2-manifold) A triangulated compact 2-manifold is a space homeomorphic to a subsetM 2 ofR n such thatM 2 is the underlying space of a simplicial complex (M 2 , T).
Example 4 The tetrahedral surface below, with triangulation
{v 0 v 1 },{v 0 v 2 },{v 0 v 3 },{v 1 v 2 },{v 1 v 3 },{v 2 v 3 }, {v 0 },{v 1 },{v 2 },{v 3 }} is a triangulated 2-manifold (homeomorphic to S 2 ).
The theorem states that every compact 2-manifold can be triangulated While the proof involves complex technical details, we will focus on the analysis of triangulated 2-manifolds It is important to note that the findings related to triangulated 2-manifolds also apply within the topological category.
Theorem 1.7 A compact,2-manifold is homeomorphic to a compact, trian- gulated 2-manifold, in other words, all compact2-manifolds are triangulable. Definitions (1-skeleton and dual 1-skeleton).
2 The dual 1-skeleton of a triangulationT equalsS
{σ j |σj is an edge of a 2-simplex in T 0 and neither vertex ofσj is a vertex of a2-simplex of
In the dual 1-skeleton, each edge connects the barycenters of two 2-simplices from the original triangulation Physically, this means that each edge consists of two segments, extending from the barycenter of a 2-simplex to the midpoint of the shared edge in the original triangulation Therefore, an edge in the dual 1-skeleton represents the combination of two 1-simplices in T 0.
Examples 5 The following are triangulable2-manifolds: a S 2 b T 2 :=S 1 ×S 1 ⊂R 4 or any other space homeomorphic to the boundary of a doughnut, the torus. c Double torus:(See Figure 1.4)
The following example cannot be embedded inR 3 ; however, it can be embed- ded in R 4 d The Klein bottle, denoted K 2 :(See Figure 1.5)
There is another 2-manifold that cannot be embedded in R 3 that we will study, which requires the use of the quotient or identification topology (seeAppendix A):
Figure 1.4: The double torus or surface of genus 2
Figure 1.5: The Klein Bottle e The projective plane, denotedRP 2 , := space of all lines through 0 in
R 3 where the basis for the topology is the collection of open cones with the cone point at the origin.
1 Show RP 2 ∼= S 2 /hx ∼ −xi, that is, the 2-sphere with diametrically opposite points identified.
2 Show that RP 2 is also homeomorphic to a disk with two edges on its boundary (called a bigon), identified as indicated in Figure 1.6.
3 Show that RP 2 ∼= M¨obius band with a disk attached to its boundary (See Figure 1.7).
Exercise 1.9 Show thatT 2 as defined above is homeomorphic to the surface in R 3 parametrized by:
There is a way of obtaining more 2-manifolds by “connecting” two or more together For instance, the double torus looks like two tori that have been joined together.
The connected sum is a mathematical operation defined for two compact, connected, triangulated 2-manifolds, denoted as M₁² and M₂² To create the connected sum, select 2-simplices D₁ and D₂ from the triangulations of M₁ and M₂, respectively Then, remove the interiors of these simplices from the manifolds, specifically M₁² - Int D₁ and M₂² - Int D₂ By pasting these modified manifolds along the boundaries of the simplices, Bd D₁ and Bd D₂, the resulting structure is known as the connected sum of M₁² and M₂², represented as M₁² # M₂² This concept can also be extended recursively to define the connected sum of multiple 2-manifolds.
This definition of connected sum can in fact be generalized to the con- nected sum of any twon-manifolds Can you see how to do it?
Exercise 1.10 Show that RP 2 #RP 2 is homeomorphic to the Klein bottle.
Exercise 1.11 Show that T 2 #RP 2 , where T 2 is the torus, is homeomor- phic K 2 #RP 2 , where K 2 is the Klein bottle.
There is another way of thinking of 2-manifolds, as the abstract spaces obtained from a particular kind of quotients (see Appendix A for a review of quotient spaces).
The process of consolidating all elements of an equivalence class into a single entity is referred to as "gluing," particularly when these equivalence classes are small, typically containing one, two, or a finite number of points.
This article explores the quotient spaces derived from polygonal disks, where interior points belong to distinct equivalence classes, edge interior points form two-point equivalence classes, and vertices are grouped into classes with varying numbers of other vertices We aim to construct a 2-manifold by pairwise gluing the edges of the polygonal disk according to a specific pattern.
In these examples, the type of arrow denotes the edges that are glued together, while the direction of the arrows illustrates the method of gluing It is essential to understand that any two gluing maps that align with the specified orientations will produce homeomorphic spaces.
Figure 1.9: Another way to see the sphere a a b b c c d d
Ensure that alternative representations of the same space are homeomorphic, and verify that these spaces are also homoeomorphic to the triangulable 2-manifolds discussed earlier.
Questions
Figure 1.13: Another version of the projective plane
The theorem will be introduced in chapter 4, presented in an equivalent form Its proof is surprisingly complex, yet its utility is undeniable.
Theorem 1.12 (Jordan Curve Theorem) Let h : [0,1]→D 2 be a topolog- ical embedding where h(0), h(1)∈Bd(D 2 ) Then h([0,1]) separates D 2 into exactly two pieces.
Theorem 1.13 Any polygonal disk with edges identified in pairs is home- omorphic to a compact, connected, triangulated 2-manifold.
Theorem 1.14 Any compact, connected, triangulated2-manifold is home- omorphic to a polygonal disk with edges identified in pairs.
The most fundamental questions in topology are:
Question 1.15 How are spaces similar and different? Particularly, which are homeomorphic? Which aren’t?
To demonstrate that two spaces are homeomorphic, we need to establish a homeomorphism between them Conversely, to prove that two spaces are not homeomorphic, we should analyze their topological properties and features This examination allows us to identify significant characteristics that can differentiate one space from another Thus, understanding these properties is crucial for distinguishing between various topological spaces.
A surface, or 2-manifold, is locally similar to R², providing insight into their local structure However, understanding the global characteristics of these spaces is essential We have encountered various examples, such as the 2-sphere, torus, Klein bottle, and RP² Our goal is to systematically categorize and classify all surfaces into distinct homeomorphism classes By leveraging the local Euclidean properties of 2-manifolds, we can better describe the overall structure of these surfaces.
When studying compact 2-manifolds, we conceptualize them as physical objects constructed from simple building blocks, specifically triangles We will focus on 2-manifolds situated in R^n that are composed of triangles This exploration is significant because every compact 2-manifold is homeomorphic to one made of a finite number of triangles embedded in R^n Utilizing a finite number of triangles allows us to employ inductive methods, facilitating the transition from one triangle to another in our analysis.
To effectively understand 2-manifolds, we should conceptualize them as tangible objects made up of a finite number of flat triangles, or simplices, that connect in specific ways These triangles can only overlap at shared edges or vertices, which provides a clear physical representation of the manifold By adopting this perspective, we can develop a reliable method for determining the overall structure of the object based on its local characteristics.
This article aims to demonstrate that every compact, triangulated 2-manifold can be constructed through the connected sum of simple 2-manifolds, specifically the sphere, torus, and projective plane, using two distinct methods.
This section presents a series of theorems demonstrating that after the removal of a single disk, any compact, triangulated 2-manifold can be understood as a disk with strips attached in straightforward configurations We consistently leverage the local structure of the triangulated 2-manifold to analyze how the entire structure integrates cohesively.
The second proof of the classification theorem views each compact, tri- angulated 2-manifold as the quotient space of a polygonal disk with its edges identified in pairs.
Definition (regular neighborhood) Let M 2 be a 2-manifold with triangu- lation T = {σ i } k i=1 Let A be a subcomplex of (M 2 , T) The regular neighborhoodof A, denoted N(A), equalsS
Exercise 2.1 explores the triangulation of a tetrahedron's boundary, represented by a triangulation T that includes four 2-simplexes, six edges, and four vertices It involves identifying the first and second derived subdivisions of T, analyzing the 1-skeleton of T, and examining the regular neighborhood of the 1-skeleton, as well as the regular neighborhoods associated with a vertex and an edge.
T, and the dual 1-skeleton ofT.
Exercise 2.2 On the accompanying pictures of the second derived sub- divisions of triangulations of the torus and the Klein bottle, find regular neighborhoods of subsets of the1-skeleton.
Exercise 2.3 Characterize graphs in the 1-skeleton of T for the triangula- tions of the sphere, torus, and projective plane whose regular neighborhoods are homeomorphic to a disk.
2.1.1 Classification of compact, connected 2-manifolds, I
This proof demonstrates that when an open disk is removed from a compact triangulated 2-manifold, the resulting space is homeomorphic to a closed disk with a certain number of bands attached to its boundary in a specific manner The classification of the surface is determined by both the number of bands and their attachment configuration.
Theorem 2.4 Let M 2 be a compact, triangulated 2-manifold with triangu- lationT Let S be a tree whose edges are 1-simplices in the 1-skeleton of T.ThenN(S), the regular neighborhood of S, is homeomorphic to D 2
Classification of 2-manifolds
PL Homeomorphism
Our objective is to categorize connected, compact triangulated 2-manifolds based on their homeomorphism type However, traditional topological homeomorphism fails to capture the intricacies of the triangulated structures associated with these manifolds To address this, we propose a method for equating two triangulated 2-manifolds that takes into account both their simplicial structure and topological type Initially, we establish an equivalence between two triangulated 2-manifolds, T1 and T2, when their simplices correspond in a clear one-to-one manner Additionally, we introduce a further equivalence concept that allows for the subdivision of the two 2-manifolds to discover new triangulations that maintain this one-to-one correspondence.
A simplicial homeomorphism between two manifolds M1 and M2, each with their respective triangulations T1 and T2, exists if there is a homeomorphism that establishes a one-to-one correspondence between the simplices of T1 and T2 This means that the homeomorphism linearly maps each simplex in T1 to a corresponding simplex in T2, ensuring that the vertices of the simplices are preserved in the mapping.
T1 go to the vertices ofT2 and the rest of the homeomorphism is determined by extending the map on the vertices linearly over each simplex.
A space can possess multiple triangulations, indicating that the notion of simplicial homeomorphism is overly restrictive For example, an underlying space with one triangulation is not simplicially isomorphic to the same space with its second derived subdivision triangulation Therefore, it is necessary to establish a broader concept of equivalence in this context.
Definition (PL homeomorphism) M 1 2 with triangulation T1 is PL home- omorphic to M 2 2 with triangulation T 2 if and only if there exist subdivisions
T 1 0 and T 2 0 of T 1 and T 2 respectively such that (M 1 2 , T 1 0 ) is simplicially iso- morphic to (M 2 2 , T 2 0 ).
The term "PL" stands for piecewise linear, indicating a homeomorphism between M 1 2 and M 2 2 that can be represented as a linear map restricted to each simplex of T 1 0 For 2-manifolds, it can be assumed that a homeomorphism between two manifolds results in a PL-homeomorphism; however, this assumption does not hold true for general n-manifolds.
Invariants
Euler characteristic
Definition (Euler characteristic) Let M 2 be a 2-manifold with triangula- tionT Let v = number of vertices in T e = number of 1-simplices in T f = number of 2-simplices in T and define the Euler characteristic,χ(M 2 ), of M 2 by χ(M 2 ) =v−e+f.
Theorem 2.25 Let M 2 be a connected, compact, triangulated 2-manifold with triangulation T Let T 0 be a subdivision of T Then χ(M 2 , T) χ(M 2 , T 0 ).
In other words, for a triangulated, compact 2-manifold, the Euler char- acteristic is preserved under subdivision.
Theorem 2.26 Let M 1 2 and M 2 2 be connected, compact, triangulated 2- manifolds If M 1 2 is PL-homeomorphic to M 2 2 , then χ(M 1 2 ) =χ(M 2 2 ).
Since PL-homeomorphic manifolds must have the same Euler character- istic, Euler characteristic helps to distinguish between 2-manifolds that are not PL-homeomorphic.
Theorem 2.28 Let M 1 2 and M 2 2 be two connected, compact, triangulated 2-manifolds Then χ(M 1 2 #M 2 2 ) =χ(M 1 2 ) +χ(M 2 2 )−2.
Theorem 2.29 Let T 2 i be the torus fori= 1, , n Then χ n
Definition (genus) The genus of S 2 = 0 The genus of a 2-manifold Σ n
Theorem 2.30 Let RP 2 i be the projective plane for i= 1, , n Then χ n
Orientability
The Euler characteristic serves as a valuable invariant for distinguishing between 2-manifolds; however, it fails to differentiate between certain surfaces, such as the torus and the Klein bottle Specifically, for each even number less than or equal to zero, there exist two non-homeomorphic compact, connected, triangulated 2-manifolds sharing the same Euler characteristic—one being a connected sum of tori and the other a connected sum of projective planes Thus, while the Euler characteristic is effective in identifying some non-homeomorphic surfaces, it does not provide a complete distinction among all surface types.
There is a second invariant which, when combined with Euler charac- teristic, will allow us to distinguish between any two non-homeomorphic, compact, connected 2-manifolds This invariant is orientability.
A surface is considered orientable if it is possible to select an ordered basis for the local Euclidean structure at each point, ensuring that these bases transition smoothly as one moves along the surface Orientability serves as a fundamental classification, categorizing 2-manifolds into two distinct groups: orientable and non-orientable Furthermore, the combination of orientability and the Euler characteristic provides a sufficient criterion to distinguish between any two compact, connected, triangulated 2-manifolds.
We can explore the concept of orientability in triangulated surfaces by considering orderings of the vertices of each simplex.
First let us see what we mean by an orientation of a 0-, 1-, and 2-simplex.
Definitions (oriented simplices) Let σ 2 be the 2-simplex {v 0 v 1 v 2 }, σ 1 be the 1-simplex{w 0 w 1 }, and σ 0 be the 0-simplex {u 0 }.
1 Two orderings of the vertices v 0 , v 1 , v n of an n-simplex σ n are said to be equivalent if they differ by an even permutation Thus {v 0 , v 1 , v 2 } ∼ {v 1 , v 2 , v 0 } However, {v 0 , v 1 , v 2 } 6∼ {v 1 , v 0 , v 2 } since they differ by a single2-cycle, which is an odd permutation Note that this equivalence relation produces precisely two equivalence classes of orderings of vertices of an n-simplex for n ≥1 An equivalence class will be denoted by [v 0 v 1 v n ], where {v 0 , v 1 , , v n } is an element of the equivalence class.
2 An orientation of the2-simplexσ 2 is a one-to-one and onto function o from the two equivalence classes of the orderings of the vertices of σ 2 to {−1,1} Note that there are two possible such orientations for σ 2 Any vertex ordering that lies in the equivalence class whose image is+1 will be called positively oriented or will be said to have a positive orientation, orderings in the other class will be said to be negatively oriented or have a negative orientation We can indicate the chosen positive orientation for σ 2 by denoting σ 2 as [v0v1v2], where [v0v1v2] is in the positive equivalence class Note that −o[v 0 v1v2] =o[v1v0v2].You can draw a circular arrow inside the 2-simplex in the direction indicated by any of the positively oriented orderings That circular arrow (which will be either clockwise or counterclockwise on the page) indicates the choice of (positive) orientation for that 2-simplex.
3 An orientationof the 1-simplex σ 1 is a one-to-one and onto function o from the two orderings of the vertices of σ 1 to{−1,1} The order- ing whose image is +1 has the positive orientation, the other has a negative orientation As with σ 2 , note that there are only two possible orientations forσ 1 , and that−o[w 0 w1] =o[w1w0] We think of[w0w1] as being the orientation that “points” from w 0 tow 1
4 Since σ 0 has a single equivalence class of orderings of its vertex, we have a slightly different definition of orientation for a 0-simplex An orientation of a0-simplex is a functiono from {[u 0 ]} to {−1,1}.
Induced orientation on an edge refers to the orientation assigned to the edges of a 2-simplex based on the chosen orientation of the simplex itself When the positive orientation of a 2-simplex is represented as [v0 v1 v2], the resulting induced orientations on the edges are as follows: a [v1 v2], b −[v0 v2] = [v2 v0], and c [v0 v1].
Induced orientation refers to a natural concept where the directed cycle of edges, connecting vertices v0 to v1 to v2 and returning to v0, aligns with a chosen positive ordering of the vertices in a 2-simplex.
Exercise 2.31 Show that the induced orientation on an edge of a2-simplex is well defined; in other words, that it is independent of the choice of positive equivalence class representative.
Definition (induced orientation on a vertex) The orientations induced on the vertices of σ 1 = [v0v1]are a −[v 0 ]. b [v1]. respectively.
An orientable, triangulated 2-manifold is defined by the ability to assign orientations to each 2-simplex such that adjacent 2-simplices maintain compatible orientations This compatibility is illustrated by the observation that when two triangles sharing an edge are both oriented counterclockwise, the shared edge will have opposing induced orientations Thus, if the orientations of the two triangles are identical, the resulting orientations on their common edge will be opposite, leading to the formal definition of orientability.
An orientable triangulated 2-manifold M2 allows for an orientation to be assigned to each 2-simplex τ in its triangulation This orientation must ensure that for any 1-simplex e that is part of the intersection of two simplices τ1 and τ2, the orientation from τ1 is opposite to that from τ2 If such an orientation cannot be established, the manifold is classified as non-orientable.
A choice of orientations of the 2-simplices of a triangulation of M 2 sat- isfying the condition stated above is called an orientation of M 2
Theorem 2.32 Suppose (M 2 , T) is a2-manifold with triangulation T and
T 0 is a subdivision of T Then if (M 2 , T) is orientable, so is (M 2 , T 0 ). Theorem 2.33 Orientability is preserved under PL homeomorphism. Theorem 2.34 M 2 is orientable if and only if it contains no M¨obius band.
Theorem 2.35 Let M =M1# .#Mn ThenM is orientable if and only ifM i is orientable for each i∈ {1, , n}.
Compact, connected, triangulated 2-manifolds are determined by ori- entability and Euler characteristic.
Theorem 2.36 (Classification of compact, connected 2-manifolds) If M 2 is a connected, compact, triangulated 2-manifold then:
(b) if M 2 is orientable and χ(M 2 ) = 2−2n, for n≥1, then
2.3 INVARIANTS 33 (c) if M 2 is non-orientable and χ(M 2 ) = 2−n, for n≥1, then
Notice that orientable connected, compact, triangulated 2-manifolds must have even Euler Characteristic.
Problem 2.37 Identify the following 2-manifolds as a sphere, or a con- nected sum ofntori (specifyingn), or a connected sum ofnprojective planes (specifyingn). a T#RP b K#RP c RP#T#K#RP d K#T#T#RP#K#T
CW complexes
Calculating the Euler Characteristic of a surface can be a tedious process, often requiring extensive triangulation Fortunately, it is not necessary to break the surface into tiny segments to accurately compute this characteristic.
We can represent a 2-manifold as a union of larger cells that fit together, allowing us to compute the Euler Characteristic Our approach to generalizing triangulation involves starting with a triangulated surface and systematically enlarging the triangles and edges to create alternative cell decompositions that still reveal the Euler Characteristic This enlargement and amalgamation process contrasts with the earlier subdivision process that preserved the Euler Characteristic, beginning with the amalgamation of two adjacent 2-simplexes from the triangulation.
Theorem 2.38 Let (M 2 , T) be a triangulated 2-manifold Suppose σ {uvw} and σ 0 = {uvw 0 } are two distinct 2-simplexes in T that share the edge e= {uv} Then we can create a new structure for M 2 alternative to
In the context of topology, let P be defined as T∪ {τ} − {σ, σ 0 , e}, where τ represents the polygon created by the union of two 2-simplices along their common edge The quantities v 0, e 0, and f 0 denote the number of vertices, edges, and polygons in P, respectively The Euler Characteristic χ(M 2, T) can then be expressed as χ(M 2, T) = v 0 − e 0 + f 0, illustrating the relationship between these elements in the geometric structure (refer to Figure 2.6).
Figure 2.6: The basic idea of CW complexes
The previous theorem amalgamated two triangles together; however, we can continue in that vein by amalgamating polygonal disks together that we may have created.
Theorem 2.39 states that for a compact, triangulated 2-manifold (M², T) with an Euler characteristic χ(M², T), we can construct a polygonal structure P inductively Starting with P₀ as T, if we have a structure Pᵢ and two 2-dimensional objects σ and σ' in Pᵢ that share a connected path of edges from vertex u to w (where v ≠ w), we can create Pᵢ₊₁ by removing σ and σ', eliminating the edges in the path from u to w, preserving only vertices u and w, and introducing the combined object σ ∪ σ' The relationship between the number of vertices (v), edges (e), and 2-dimensional objects (f) in Pᵢ₊₁ is given by the formula χ(M², T) = v - e + f, as illustrated in Figure 2.7.
Figure 2.7: Removing a path from a CW complex
In a 2-dimensional space in P i, a 2-dimensional object may not be homeomorphic to a disk; however, the 'interior' of each object remains homeomorphic to an open disk We can extend our inductive definition of the new structure on M 2 by progressively reducing the number of 1-dimensional objects.
Theorem 2.40 states that for a compact, triangulated 2-manifold (M², T) with an inductively defined polygonal structure P, we can replace P with a new structure Starting with P = P0, if any configuration Pi contains an edge e with a free vertex v—meaning v does not belong to the boundary of any other edge in Pi—we can eliminate both v and e from Pi to form the new structure.
In the process of constructing the sequence P i+1, if a vertex v in P i serves as an endpoint for both an edge e and another edge f, and is not connected to any other edges, then we can remove v, e, and f from P i This allows us to introduce a new 1-dimensional object formed by the union of e and f, resulting in the updated structure P i+1 Furthermore, if v 0, e 0, and f 0 represent the counts of vertices, 1-dimensional objects, and 2-dimensional objects in the inductively defined structure P, the relationship χ(M 2, T) can be expressed as χ(M 2, T) = v 0 - e 0 + f 0.
Exercise 2.41 Start with a triangulation ofS 2 and carry out the preceding process as far as possible What “structure” do you get? Confirm that you get the right Euler Characteristic.
Exercise 2.42 Start with a triangulation ofT 2 and carry out the preceding process as far as possible What “structure” do you get? Confirm that you get the right Euler characteristic.
Figure 2.8: Removing edges, vertices, and faces from a CW complex
We will now formalize what we have observed by defining a CW decom- position of a 2-manifold.
Definition (interior of a 0−, 1−, and 2-simplex) 1 For each2-simplex σ 2 ={v 0 v1v2} let Intσ 2 ={λ 0 v0+λ1v1+λ0v2|0< λi