Topology – Page 2 – Mathtuition88

Motivation of Simplicial Sets

A simplicial set is a purely algebraic model representing topological spaces that can be built up from simplices and their incidence relations. This is similar to the method of CW complexes to modeling topological spaces, with the critical difference that simplicial sets are purely algebraic and do not carry any actual topology.

To return back to topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. A topological space $X$ is said to be compactly generated if it satisfies the condition: A subspace $A$ is closed in $X$ if and only if $A\cap K$ is closed in $K$ for all compact subspaces $K\subseteq X$ . A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff.

Homotopy Theory on Simplicial Sets

Let $f,g:X\to Y$ be simplicial maps. We say that $f$ is homotopic to $g$ (denoted by $f\simeq g$ ) if there exists a simplicial map $F:X\times I\to Y$ such that $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x\in X$ . If $A$ is a simplicial subset of $X$ and $f,g:X\to Y$ are simplicial maps such that $f|_A=g|_A$ , we say that $f\simeq g\ \text{rel}\ A$ if there is a homotopy $F:X\times I\to Y$ such that $F(x,0)=f(x)$ , $F(x,1)=g(x)$ and $F(a,t)=f(a)$ for all $x\in X$ , $a\in A$ , $t\in I$ .

Let $X$ be a simplicial set. The elements $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ are said to be matching faces with respect to $i$ if $d_jx_k=d_kx_{j+1}$ for $j\geq k$ and $k,j+1\neq i$ .

A simplicial set $X$ is said to be fibrant (or Kan complex) if it satisfies the following homotopy extension condition for each $i$ :

Let $x_0,\dots,x_{i-1},x_{i+1},\dots,x_n\in X_{n-1}$ be any elements that are matching faces with respect to $i$ . Then there exists an element $w\in X_n$ such that $d_jw=x_j$ for $j\neq i$ .

We define $\Lambda^i[n]$ as the simplicial subset of $\Delta[n]$ generated by all $d_j\sigma_n$ for $j\neq i$ , where $\sigma_n=(0,1,\dots\,n)\in\Delta[n]_n$ is the nondegenerate element.

Proposition: Let $X$ be a simplicial set. Then $X$ is fibrant if and only if every simplicial map $f:\Lambda^i[n]\to X$ has an extension for each $i$ .

Wedge and Smash Products

Let $X$ and $Y$ be pointed simplicial sets. The wedge of $X$ and $Y$ , denoted by $X\vee Y$ , is the simplicial set obtained by identifying the basepoint of $X$ with the basepoint of $Y$ . Hence, we can define $X\vee Y$ by the push-out diagram
wedge smash
$X\vee Y$ can be described as the simplicial subset of $X\times Y$ consisting of $(x,*)$ for $x\in X$ and $(*,y)$ for $y\in Y$ .

The smash product $X\wedge Y$ is defined to be the simplicial quotient $X\times Y/(X\vee Y)$ .

Push-out

Let $f:A\to B$ and $g:A\to C$ be simplicial maps. We define $X_n$ to be the push-out in the diagram
Screen Shot 2016-01-17 at 4.46.49 PM
i.e. $X_n=B_n\coprod C_n/\sim$ , where $\sim$ is the equivalence relation such that $f(a)\sim g(a)$ for $a\in A_n$ . Then $X=\{X_n\}_{n\geq 0}$ with faces and degeneracies induced from that in $B$ and $C$ forms a simplicial set with a push-out diagram of simplicial sets
Screen Shot 2016-01-17 at 4.47.03 PM
For example, let $\partial\Delta[n]=\langle d_i(0,1,\dots,n)\mid 0\leq i\leq n\rangle\subseteq \Delta[n]$ be the simplicial subset of $\Delta[n]$ generated by all of the faces of the $n$ -simplex $\sigma_n=(0,1,\dots,n)$ . Then

is a push-out diagram, where $f$ is the inclusion map and $S^n=\Delta[n]/\partial\Delta[n]$ .

The Shape of Space

Just came across this book: The Shape of Space (Chapman & Hall/CRC Pure and Applied Mathematics). It is a very unique book, in the sense that it is aimed at high school students, but even a undergraduate or graduate student can benefit from it. It has a lot of diagrams, that are missing in most textbooks, presumably because it takes a lot of effort to draw a mathematical (3D) diagram.

It will be useful to students who want to learn more about topology. This book can be read casually, it is not like a textbook, yet it has substantial mathematical content.

Example of an illustration in the book:

illustration topology

Behavior of Homotopy Groups with respect to Products

This blog post is on the behavior of homotopy groups with respect to products. Proposition 4.2 of Hatcher:

For a product $\prod_\alpha X_\alpha$ of an arbitrary collection of path-connected spaces $X_\alpha$ there are isomorphisms $\pi_n(\prod_\alpha X_\alpha)\cong\prod_\alpha \pi_n(X_\alpha)$ for all $n$ .

The proof given in Hatcher is a short one: A map $f:Y\to \prod_\alpha X_\alpha$ is the same thing as a collection of maps $f_\alpha: Y\to X_\alpha$ . Taking $Y$ to be $S^n$ and $S^n\times I$ gives the result.

A possible alternative proof is to first prove that $\pi_n(X_1\times X_2)\cong\pi_n(X_1)\times\pi_n(X_2)$ , which is the result for a product of two spaces. The general result then follows by induction.

We construct a map $\psi:\pi_n(X_1\times X_2)\to\pi_n(X_1)\times\pi_n(X_2)$ , $\psi([f])=([f_1],[f_2])$ .

Notation: $f:S^n\to X_1\times X_2$ , $f_1=p_1\circ f:S^n\to X_1$ , $f_2=p_2\circ f:S^n\to X_2$ where $p_i:X_1\times X_2\to X_i$ are the projection maps.

We can show that $\psi ([f]+[g])=\psi([f])+\psi([g])$ , thus $\psi$ is a homomorphism.

We can also show that $\psi$ is bijective by constructing an explicit inverse, namely $\phi:\pi_n(X_1)\times\pi_n(X_2)\to\pi_n(X_1\times X_2)$ , $\phi([g_1],[g_2])=[g]$ where $g:S^n\to X_1\times X_2$ , $g(x)=(g_1(x),g_2(x))$ .

Thus $\psi$ is an isomorphism.

The Fundamental Group

Source: Topology (2nd Economy Edition)

If we pick a point $x_0$ of the space X to serve as a “base point” and consider only those paths that begin and end at $x_0$ , the set of these path-homotopy classes does form a group under *. It will be called the fundamental group of X.

The important thing about the fundamental group is that it is a topological invariant of the space X, and it will be crucial in studying homeomorphism problems.

Definition of fundamental group:
Let X be a space; let $x_0$ be a point of X. A path in X that begins and ends at $x_0$ is called a loop based at $x_0$ . The set of path homotopy classes of loops based at $x_0$ , with operation *, is defined as the fundamental group of X relative to the base point $x_0$ . It is denoted by $\pi_1 (X,x_0)$ .

Previously, we have shown that the operation * satisfies the axioms for a group. (See our earlier blog posts on the associativity properties and other groupoid properties of the operation.

This group is called the first homotopy group of X. There is also a second homotopy group, and even groups $\pi_n (X,x_0)$ for all $n\in \mathbb{Z}^+$ .

An example of a fundamental group:

$\pi_1 (\mathbb{R}^n,x_0)$ is the trivial group (the group consisting of just the identity). This is because if f is a loop in $\mathbb{R}^n$ based at $x_0$ , the straight line homotopy is a path homotopy between f and the constant path at $x_0$ .

An interesting question (discussed in the next upcoming blog posts) would be how the group depends on the base point $x_0$ .

Proof of Associativity of Operation * on Path-homotopy Classes

(Continued from https://mathtuition88.com/2015/06/25/the-groupoid-properties-of-operation-on-path-homotopy-classes-proof/)

Earlier we have proved the properties (2) Right and left identities, (3) Inverse, leaving us with (1) Associativity to prove.

For this proof, it will be convenient to describe the product f*g in the language of positive linear maps.

First we will need to define what is a positive linear map. We will elaborate more on this since Munkres’ books only discusses it briefly.

Definition: If [a,b] and [c,d] are two intervals in $\mathbb{R}$ , there is a unique map $p:[a,b]\to [c.d]$ of the form p(x)=mx+k that maps a to c and b to d. This is called the positive linear map of [a,b] to [c,d] because its graph is a straight line with positive slope.

Why is it a positive slope? (Not mentioned in the book) It turns out to be because we have:

p(a) = ma+k=c

p(b) = mb+k=d

Hence, d-c = mb-ma = m(b-a)

Thus, m=(d-c)/(b-a), which is positive since d-c and b-a are all positive quantities.

Note that the inverse of a positive linear map is also a positive linear map, and the composite of two such maps is also a positive linear map.

Now, we can show that the product f*g can be described as follows: On [0,1/2], it is the positive linear map of [0,1/2] to [0,1], followed by f; and on [1/2,1] it equals the positive linear map of [1/2,1] to [0,1], followed by g.

Let’s see why this is true. The positive linear map of [0,1/2] to [0,1] is p(x)=2x. fp(x) = f(2x).

The positive linear map of [1/2,1] to [0,1] is p(x)=2x-1. gp(x)=g(2x-1).

If we look back at the earlier definition of f*g, that is precisely it!

Now, given paths, f, g, and h in X, the products f*(g*h) and (f*g)*h are defined if and only if f(1)=g(0) and g(1)=h(0), i.e. the end point of f = start point of g, and the end point of g = start point of h. If we assume that these two conditions hold, we can also define a triple product of the paths f, g, and h as follows:

Choose points a and b of I so that 0<a<b<1. Define a path $k_{a,b}$ in X as follows: On [0,a] it equals the positive linear map of [0,a] to I=[0,1] followed by f; on [a,b] it equals the positive linear map of [a,b] to I followed by g; on [b,1] it equals the positive linear map of [b,1] to I followed by h. This path $k_{a,b}$ depends on the choice of the values of a and b, but its path-homotopy class turns out to be independent of a and b.

We can show that if c and d are another pair of points of I with 0<c<d<1, then $k_{c,d}$ is path homotopic to $k_{a,b}$ .

Let $p:I\to I$ be the map whose graph is pictured in Figure 51.9 (taken from Munkre’s Book)

On the intervals [0,a], [a,b], [b,1], it equals the positive linear maps of these intervals onto [0,c],[c,d],[d,1] respectively. It follows that $k_{c,d} \circ p = k_{a,b}$ . Let’s see why this is so.

On [0,a] $k_{c,d}\circ p$ is the positive linear map of [0,a] to [0,c], followed by the positive linear map of [0,c] to I, followed by f. This equals the positive linear map of [0,a] to I, followed by f, which is precisely $k_{a,b}$ . Similar logic holds for the intervals [a,b] and [b,1].

$p$ is a path in I from 0 to 1, and so is the identity map $i: I\to I$ . Since I is convex, there is a path homotopy P in I between p and i. Then, $k_{c,d}\circ P$ is a path homotopy in X between $k_{a,b}$ and $k_{c.d}$ .

Now the question many will be asking is: What has this got to do with associativity. According to the author Munkres, “a great deal”! We check that the product $f*(g*h)$ is exactly the triple product $k_{a,b}$ in the case where $a=1/2$ and $b=3/4$ .

By definition,

$(g*h)(s)=\begin{cases} g(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ h(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases}$

Thus, $f*(g*h)(s)=\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ (g*h)(2s-1)\ &\text{for }s\in [\frac{1}{2},1] \end{cases} =\begin{cases} f(2s)\ &\text{for }s\in [0,\frac{1}{2}]\\ g(4s-2)\ &\text{for }s\in [\frac{1}{2},\frac{3}{4}]\\ h(4s-3) &\text{for }s\in [\frac{3}{4},1] \end{cases}$

We can also check in a very similar way that $(f*g)*h)=k_{c,d}$ when c=1/4 and d=1/2. Thus, the these two products are path homotopic, and we have finally proven the associativity of *.

Reference:

Topology (2nd Economy Edition)

The Groupoid Properties of * on Path-homotopy Classes

This is one of the first instances where algebra starts to appear in Topology. We will continue our discussion of material found in Topology (2nd Economy Edition) by James R. Munkres.

First, we need to define the binary operation *, that will later make * satisfy properties that are very similar to axioms for a group.

Definition: If f is a path in X from $x_0$ to $x_1$ , and if g is a path in X from $x_1$ to $x_2$ , we define the product f*g of f and g to be he path h given by the equations

$h(s)=\begin{cases}f(2s) &\text{for }s\in [0,\frac{1}{2}], \\ g(2s-1)& \text{for }s\in[\frac{1}{2}, 1]\end{cases}$

Well-defined: The function h is well-defined, at s=1/2, f(1)=x_1, g(0)=x_1.

Continuity: h is also continuous by the pasting lemma.

h is a path in X from x_0 to x_2. We think of h as the path whose first half is the path f and whose second half is the path g.

We will verify that the product operation on paths induces a well-defined operation on path-homotopy classes, defined by the equation $[f]*[g]=[f*g]$

Let F be a path homotopy between f and f’, and let G be a path homotopy between g and g’.

i.e. we have F(s,0)=f(s), F(s,1)=f'(s)
F(0,t)=x_0, F(1,t)=x_1
G(s,0)=g(s), G(s,1)=g'(s)
G(0,t)=x_1, G(1,t)=x_2

We can define:

$H(s,t)=\begin{cases}F(2s,t) &\text{for }s\in[0,\frac{1}{2}],\\ G(2s-1,t)&\text{for }s\in[\frac{1}{2},1]\end{cases}$ .

We can check that F(1,t)=x_1=G(0,t) for all t, hence the map H is well-defined. H is continuous by the pasting lemma.

Let’s check that H is the required path homotopy between f*g and f’*g’.

For s in [0,1/2],

H(s,0) = F(2s,0) =f(2s)=h(s)

H(s,1) = F(2s,1) =f'(2s)=h'(s)
h’ := f’ * g’

H(0,t) = F(0,t) = x_0

s in [1/2,1] works fine too:

H(s,0) = G(2s-1,0) = g(2s-1)=h(s)

H(s,1) = G(2s-1,1)= g'(2s-1) = h'(s)

H(1,t) = G(1,t)= x_2

Thus, H is indeed the required path homotopy between f*g and f’*g’. * is almost like a binary operation for a group. The only difference is that [f]*[g] is not defined for every pair of classes, but only for those pairs [f], [g] for which f(1) = g(0), i.e. the end point of f is the starting point of g.

Book Review: Topology, James R. Munkres

Topology (2nd Edition)

This book is the best introductory book on Topology, an upper undergraduate/graduate course taken in university. I have written a short book review on it.

Excerpt:

Book Review: Topology
Book’s Author: James R. Munkres
Title: Topology
Prentice Hall, Second Edition, 2000

It is often said that one must not judge a book by its cover. The book with a plain cover, simply titled “Topology”, is truly a rare gem and in a class of its own among Topology books.

One striking aspect of the book is that it is almost entirely self-contained. As stated in the preface, there are no formal subject matter prerequisites for studying most of the book. The author begins with a chapter on Set Theory and Logic which covers necessary concepts like DeMorgan’s laws, Countable and Uncountable Sets, and the Axiom of Choice.

The first part of the book is on General Topology. The second part of the book is on Algebraic Topology. The book covers Topological Spaces and Continuous Functions, Connectedness and Compactness, and Separation Axioms. Some other material in the book include the Tychonoff Theorem, Metrization Theorems and Paracompactness, Complete Metric Spaces and Function Spaces, and Baire Spaces and Dimension Theory.

The book defines connectedness as follows: The space $X$ is said to be connected if there does not exist a separation of $X$ . (A separation of $X$ is defined to be a pair $U$ , $V$ of disjoint nonempty open subsets of $X$ whose union is $X$ .) Other sources may define connectedness by, $X$ is connected if $\nexists$ continuous $f:X\twoheadrightarrow \mathbf{2}$ .

Also, the proof of Urysohn’s Lemma in the book was presented slightly differently from other books as they did not use dyadic rationals to index the family of open sets. Rather, the book lets $P$ be the set of all rational numbers in the interval $[0,1]$ , and since $P$ is countable, one can use induction to define the open sets $U_p$ . In hindsight, the dyadic rationals approach in other sources may be more explicit and clearer.

An interesting new concept mentioned in the book is that of locally connectedness (not to be confused with locally path connectedness). A space $X$ is said to be locally connected at $x$ if for every neighborhood $U$ of $x$ , there is a connected neighborhood $V$ of $x$ contained in $U$ . If $X$ is locally connected at each of its points, it is said simply to be locally connected. For example, the subspace $[-1,0)\cup (0,1]$ of $\mathbb{R}$ is not connected, but it is locally connected. The topologists’ sine curve is connected but not locally connected.

In general, the content of the book is comprehensive. The other book, “Essential Topology”, did not cover some topics like the Urysohn Lemma, regular spaces and normal spaces.

Approach
The author’s approach is generally to give a short motivation of the concept, followed by definitions and then theorems and proofs. Examples are interspersed in between the text. The motivation tends to be a little bit too short though. For instance, in other books there is some motivation of how balls can determine the metric in a metric space, leading to the concepts of “candidate balls” $\mathcal{C}=\{C_\epsilon (x)\}_{\epsilon >0, x\in X}$ . This useful concept is not found in the book Topology, nor the other book Essential Topology.

One interesting explanation of the terminology “finer” and “coarser” is found in the book. The idea is that a topological space is like “a truckload full of gravel”‘ — the pebbles and all unions of collections of pebbles being the open sets. If now we smash the pebbles into smaller ones, the collection of open sets has been enlarged, and the topology, like the gravel, is said to have been made finer by the operation.

Another point to note is that the book does not use Category Theory. Personally, I would prefer the Category approach, since it can make proofs neater, and it provides additional insight to the nature of the theorem. We also note that the other book “Essential Topology”, also does not explicitly use Category Theory. But upon closer examination, the book has expressed commutative diagrams in words, which is not as clear as in diagram form.

Organization
The organization of the book is similar to most other books, except that it covers Connectedness and Compactness before the Separation Axioms. The concept of Hausdorff spaces, however, is covered way earlier, immediately after the discussion of closure and interior of a set. This enables theorems like “Every compact subspace of a Hausdorff space is closed” to be proved in the Compactness chapter.

Style
The author’s style is to combine rigor in proofs and definitions, with intuitive ideas in the examples and commentary. This makes it both a good textbook to learn from, and a good reference for proofs too.

This informal style in the commentary makes for a especially good read. For instance, a mathematical riddle is mentioned: “How is a set different from a door?” (For interested readers, the answer can be found on page 93.)

Also, there are many figures in the book, 84 sets of figures to be precise. This is rather good for a math book, and I would recommend the book to visual learners.

However, to learn Topology from this book alone may be difficult. Even though there are exercises to practice, there are no solutions and very few hints. Also, the book uses the terminology “limit point”, which can be confusing.

The book has surprisingly few typographical errors. While reading through the book, I only spotted a trivial one on page 107, where a function written as “ $F$ ” should be “ $f$ ” instead. Upon consulting an errata list, there was only one page of errors.

Conclusion
In conclusion, despite some shortcomings of the book, Topology is a great book, and if there was one Topology book that I could bring to a desert island, it would be this one.

Chapter Headings

Part I: General Topology

Set Theory and Logic
Topological Spaces and Continuous Functions
Connectedness and Compactness
Countability and Separation Axioms
The Tychonoff Theorem
Metrization Theorems and Paracompactness
Complete Metric Spaces and Function Spaces
Baire Spaces and Dimension Theory

Part II: Algebraic Topology

The Fundamental Group
Separation Theorems in the Plane
The Seifert-van Kampen Theorem
Classification of Surfaces
Classification of Covering Spaces
Applications to Group Theory

For more undergraduate Math book recommendations, check out Undergraduate Level Math Book Recommendations.