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{\huge Peter Saveliev\par} %\scshape \bfseries

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{\Huge\bfseries Topology Illustrated\par}

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{\large With 1000 Illustrations\par} %\itshape

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\begin{flushleft}

Peter Saveliev\\

Department of Mathematics\\

Marshall University\\

Huntington, WV 25755\\

USA\\

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Topology Illustrated by Peter Saveliev\\

670 pages, includes index\\

ISBN 978-1-4951-8875-6\\

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Mathematics Subject Classification (2010): 55-01, 57-01, 54-01, 58-01, 39A12\\

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{\Large\itshape Dedicated to the memory of my parents\par}

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\section*{Preface}

A first course in topology is usually a semester in point-set topology. Sometimes a chapter on the fundamental group is included at the end, with very little time left. For the student, algebraic topology often never comes.

The main part of the present text grew from the course ''Topology I and II'' that I have taught at Marshall University in recent years. This material follows a two-semester first course in topology with emphasis on algebraic topology. Some common topics are missing, though: the fundamental group, classification of surfaces, and knots. Point-set topology is presented only to the degree that seems necessary in order to develop algebraic topology; the rest is likely to appear in a, typically required, real analysis course. The focus is on ''homology''.

Such tools of algebraic topology as chains and cochains form a foundation of ''discrete calculus''. A through introductionis provided.

The presentation is often more detailed than one normally sees in a textbook on this subject which makes the text useful for self-study or as a companion source.

There are over 1000 exercises. They appear just as new material is being developed. Some of them are quite straight-forward; their purpose is to slow you down.

There are over 1000 pictures. They are used -- only as ''metaphors'' -- to illustrate topological ideas and constructions. When a picture is used to illustrate a proof, the proof still remains complete without it.

Applications are present throughout the book. However, they are neither especially realistic nor (with the exception of a few spreadsheets to be found on the author's website) computational in nature; they are mere ''illustrations'' of the topological ideas. Some of the topics are: the shape of the universe, configuration spaces, digital image analysis, data analysis, social choice, exchange economy, and, of course, anything related to calculus. As the core content is independent of the applications, the instructor can pick and choose what to include.

Both pictures and exercises are spread evenly through the sections (average: $1.5$ per page).

The way the ideas are developed may be called “historical”, but not in the sense of what actually happened -- it's been too messy -- but rather what ''ought to'' have happened.

All of this makes the book a lot longer than a typical book with a comparable coverage. Don't be discouraged!

A rigorous course in linear algebra is an absolute necessity. In the early chapters, one may be able to avoid the need for a modern algebra course but not the maturity it requires.

$\hspace*{5 mm}\bullet$ {\bf Chapter I: Cycles} contains an informal introduction to homology as well as a sample: homology of graphs. \\

$\hspace*{5 mm}\bullet$ {\bf Chapter II: Topologies} is the starting point of point-set topology, developed as much as is needed for the next chapter. \\

$\hspace*{5 mm}\bullet$ {\bf Chapter III: Complexes} introduces first cubical complexes, cubical chains, unoriented and then oriented, and cubical homology. Then a thorough introduction to simplicial homology is provided. \\

$\hspace*{5 mm}\bullet$ {\bf Chapter IV: Spaces} continues to build the necessary concepts of point-set topology and uses them to develop further ideas of algebraic topology, such as homotopy and cell complexes. \\

$\hspace*{5 mm}\bullet$ {\bf Chapter V: Maps} presents homology theory of polyhedra and their maps. \\

$\hspace*{5 mm}\bullet$ {\bf Chapter VI: Forms} introduces discrete differential forms as cochains, calculus of forms, cohomology, and metric tensor on cell complexes.\\

$\hspace*{5 mm}\bullet$ {\bf Chapter VII: Flows} applications of calculus of forms to ODEs, PDEs, and social choice.

By the end of the first semester, one is expected to reach the section on simplicial homology in Chapter III, but maybe not the section on the homology maps yet. For a single-semester first course, one might try this sequence: Chapter II, Sections III.4 - III.6, Chapter IV (except IV.3), Section V.1. For a one-semester course that follows point-set topology (and modern algebra), one can take an accelerated route: Chapters III - V (skipping the applications). For discrete calculus, follow: Sections III.1 - III.3, Chapters VI and VII.

The book is mostly undergraduate; it takes the student to the point where the tough proofs are about to start to become unavoidable. Where the book leaves off, one usually proceeds to such topics as: the axioms of homology, singular homology, products, homotopy groups, or homological algebra. ''Geometry and Topology'' by Bredon is a good choice.

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\begin{flushright} Peter Saveliev \end{flushright}

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\includegraphics[height=22mm,width=14cm,keepaspectratio]{Escher and I.png}

About the cover. Once upon a time I took a better look at the poster of ''Drawing Hands'' by Escher hanging in my office and realized that what it depictslooks symmetric but isn't! I decided to fix the problem and the result is called ''Painting Hands''. This juxtaposition illustrates how the antipodal map (aka the central symmetry) reverses orientation in the odd dimensions and preserves it in the even dimensions. That's why to be symmetric the original would have to have two right hands!

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\chapter{Cycles}

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\section{Topology around us}

==Topology -- Algebra -- Geometry==

In an attempt to capture the essence of topology in a single sentence, we usually say that ''topology is the science of spatial properties that are preserved under continuous transformations.'' To elaborate a bit: you can bend, stretch, and shrink but '''not''' tear or glue.

In order to illustrate this idea in the context of mathematics as a whole, let's take a look at these delicious donuts:

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{\bf TOPOLOGY: }

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Count: {\it One, two, three, four... five!}

{\bf ALGEBRA: }

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Compute: {\it Two times three... six!}

{\bf GEOMETRY: }

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Measure: {\it Seven... eighths!}

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In order to see how topology is necessary for counting, consider the fact that the first step is to recognize that these are separate objects -- disconnected from each other! Furthermore, to count the holes, we need to recognize them as a different kind of topological feature. In fact, counting, computing, and measuring are all preceded by our ability to perceive the topology in these pictures.

As we shall see, ''topology is the science of spatial properties that don't rely on measuring''.

Now, to answer “why do we need to study topology?”, we start with a few elementary examples. They come from four seemingly unrelated areas: vision and computer vision, cosmology, data analysis, and social choice theory.

==The integrity of everyday objects==

In order to be able to delegate some of the decision making to computers, one has to start with what he intuitively understands but try to describe it in absolutely unambiguous terms.

In industrial context, one might need to consider the integrity of objects being manufactured or inspected.

The first question may be: this bolt is supposed to hold two things together; is it still capable of that, or ''is there a crack in it''?

[[Image:bolt broken.png|center]]

In other words: would the bolt hold a ''hair'' or would the hair slip through?

The second question may be: this porous material is supposed to stop a flow of liquid; is it still water-tight or ''is there leakage''?

[[Image:Sponge.png|center]]

In other words: would the sheet hold ''water'' or might some permeate through?

The third question may be: to be strong, this alloy is supposed to be solid; is it still solid, or ''are there air bubbles''?

[[Image:bubbles in metal.png|center]]

In other words: would the alloy hold no ''air'', or might some get in?

It is important to understand now that these are three ''different'' kinds of integrity loss -- as there may be a crack but no hole or vice versa, etc.:

[[image:different kinds of integrity loss.png|center]]

We can describe the three situations informally as follows. The objects have:

*''cuts'', or

*''tunnels'', or

*''voids''.

'''Exercise.''' Sketch examples for all possible combinations of cuts, tunnels, and voids with one or none of each and indicate corresponding real-life objects.

Of course, the presence of one of these features doesn't have to mean that the object is defective, as in the examples: a rope, a bucket, and paint. The examples of objects that are ''intended'' to have such a topological feature are respectively: sliced bread, a strainer, and a balloon.

[[image:different kinds of integrity features intended.png|center]]

Next, we will classify these three types of “defects”, according to their ''dimensions''.

To understand why and how, let's recall from the elementary geometry (or linear algebra) this list

$0$-dimensional, $1$-dimensional and $2$-dimensional spaces:

*0. points,

*1. lines,

*2. planes.

[[image:points, lines, and planes.png|center]]

If these spaces are allowed to be deformed, the list becomes:

*0. points,

*1. curves,

*2. surfaces.

[[image:points, curves, and surfaces.png|center]]

Some of those are special -- in the sense that they have no end-points or edges, i.e., there are no ''boundaries'', as in these:

*0. point,

*1. circle,

*2. sphere.

[[image:points, curves, and surfaces as cycles.png|center]]

These,as well as their deformed versions, are called ''cycles''. Meanwhile, objects that have no such features are called ''acyclic''.

Finally, our conclusions.

$\bullet$ 0. Any two points form a $0$-cycle and, since this is the simplest example of a ''cut'', the latter is a $0$-dimensional feature:

[[image:two points cut apart.png|center]]

$\bullet$ 1. Any circle is a $1$-cycle and, since this is the simplest example of a ''tunnel'', the latter is a $1$-dimensional feature:

[[image:looking through circle.png|center]]

$\bullet$ 2. Any sphere is a $2$-cycle and, since this is the simplest example of a ''void'', the latter is a $2$-dimensional feature:

[[image:head inside sphere.png|center]]

'''Exercise.''' Using this terminology, describe the topology of: a basketball, a football, a cannonball, a doughnut, an inner tire, a steering wheel, a bicycle wheel, a potter's wheel, a fingerprint, a tree, an envelope.

'''Exercise.''' Suggest your own examples of topological issues in everyday life and describe them using this terminology.

What if we ''deform'' these objects as if they are made of rubber? We can stretch and shrink them and, as long as we do not cut or glue, the number of topological features will remain the same. Indeed, the cuts, holes, and voids may narrow but won't disappear. Indentations may appear but they won't turn into topological features.

This property is exemplified by an amoeba -- a single-cell organism able to freely change its form.

[[image:amoeba.png|center]]

'''Exercise.''' Sketch a sequence of steps to show how this man -- let us first appoint him amoeba-like abilities -- can unlock his hands while his fingers remain together and continue to form the two loops.

[[image:amoeba hands.png|center]]

What about more general ''continuous transformations''?

Breaking a bolt is not continuous but welding it back together is. Digging a tunnel (all the way) through a wall is not continuous but filling it shut is. Piercing a bubble is not continuous but patching it is. Bread is cut, tires are punctured, paper is folded into an origami, fabric is sewed into a suit or an airbag, etc., etc.

As these examples show, even ''continuous transformations'' can create and remove topological features.

'''Exercise.''' Describe what happens to the three topological features in the above examples.

'''Exercise.''' Describe, in precise terms, what happens to the amoeba's topology as it feeds? Indicate which stages of development are continuous and which aren't.

==The shape of the Universe==

What is the shape of the Universe? What is its topology? Does the Universe have “cuts”, “tunnels”, or “voids”?

Looking around, we don't observe any of these! But remember how, hundreds of years ago, sailors started to notice the curvature of the horizon? And later they ''proved'' -- by around-the-world travel and even later by orbiting -- that the surface of the Earth encloses something inside?

As for the Universe, what we ''know'' is that we are not living in the flat world of Euclid and Newton; we are living in the curved world of Riemann and Einstein:

[[Image:curved universe.png|center]]

But the Universe that curves and bends ''here'' might curve and bend ''everywhere!'' Then, no matter how small the bend is, it might ''close on itself'':

[[image:bent universe is cyclic.png|center]]

It may be possible then to travel in a straight line and arrive at the starting point from the opposite direction. In fact, the light of the Sun may come to us from an unexpected direction and, as a result, appear to come from a distant star:

[[Image:Sun and you.png|center]]

Events of this kind would provide us with clues about the topology of the Universe. We have to rely on such indirect observations because we can't step outside and observe the Universe from there:

[[Image:you_and_god.png|center]]

As we stay ''inside'' our Universe, we can't see its “cuts”, “tunnels”, or “voids”. The only way we can discover its structure is by traveling or by observing others -- such as light or radio waves -- ''travel''.

If multiple universes exist but we have no way of traveling there, we can't confirm their existence. So much of $0$-cycles...

Traveling in various directions and coming (or not coming) back will produce information about loops in space. These loops, or $1$-cycles, are used to detect tunnels in the Universe.

There might also be voids and, since the Universe is $3$-dimensional, it might also have $3$-dimensional topological features. Studying these features would require us to add to the existing list, currently containing the point, the line, and the plane, a new item: space, or, more accurately: a $3$-dimensional space. How such a space creates a $3$-''cycle'' may be hard or impossible to visualize. Nonetheless, these cycles '''are''' detectable--once we arm ourselves with the methods presented in this book.

'''Example.''' What if the Universe is like a room with two doors and, as you exit through one door, you enter it through the other? You'll realize that, in fact, it's the same door! If you look through this doorway, this is what you see:

[[image:view through quotient door.png|center]]

$\hspace\fill\square$

'''Exercise.''' What is the dimension of this feature?

'''Exercise.''' What if, as you exit one door, you enter the other -- but upside down? Sketch what you would see.

Elsewhere, we face even higher dimensions...

==Patterns in data==

Data resides outside of our tangible, physical, $3$-dimensional world.

Imagine we conduct an ''experiment'' that consists of a set of $100$ different measurements. We would, of course, like to make sense of the results. First, we put the experiment's results in the form of a string of $100$ numbers. Next, thinking mathematically, we see this string as a point in the $100$-dimensional space. Now, if we repeat the experiment $1000$ times, the result is a “point cloud” of $1000$ points in this space:

[[Image:point cloud of the plane.png|center]]

Now, as scientists, we are after ''patterns'' in data. So, is there a pattern in this point cloud and maybe a law of nature that has produced it? What shape is hinted by the data?

[[Image:point clouds.png|center]]

But our ability to see is limited to three dimensions!

'''Exercise.''' What is the highest dimension of data your spreadsheet software can visualize? Hint: don't forget about colors.

With this limited ability to visualize, we still need to answer the same questions about the shape outlined by the point cloud:

*Is it one piece or more?

*Is there a tunnel or a void?

*And what about possible $100$-dimensional topological features?$\\$

Once again, we ''can't'' step outside and take a look at this object.

The first question is the most important as it is the question of ''classification''. We assign to classes what we see: drugs, customers, movies, species, web sites, etc. The methods for solving this problem -- often called “clustering” -- are well-developed in data analysis:

[[image:three clusters.png|center]]

The rest of the questions will require topological thinking presented in this book. We will record cycles of all dimensions in the data and, through ''purely algebraic'' procedures, detect its topological features: