I wrote my Master’s thesis.

# Category: Topology

## Notes on One-Parameter Deformations of Cohomology Theories

*Thanks to Dr. Lubin for graciously helping me derive and understand one-parameter families of formal group laws, thanks to Eric Peterson for introducing me to Morava’s Forms of K-theory, thanks to Agnes Beaudry for pointing out that there was a neater way to check for Landweber-exactness.*

Today, I want to discuss the opposite procedures of deformations and contractions of complex-orientable cohomology theories.

Really, all I want is to illustrate the fact that, with the appropriate combinations of both procedures, we obtain new cohomology theories. So, we’re going to examine the construction of one-parameter families of cohomology theories via one-parameter deformations of formal group laws — of particular interest is the case where a continuous deformation causes an increase of chromatic height.

#### Motivation and Story Leading To This Construction

I was playing with the concept of group contraction, and thinking about the construction of elliptic cohomology theories.

I accidentally constructed a model of Morava E-theory of height 2 at the prime 3. (We didn’t realize that it was \(E_{(2)}(3)\) at first, we just thought it was some weird cohomology theory.)

I found this kind of enlightening and so I want to show you how I came across it.

Morava constructed a family of elliptic cohomology theories by deforming K-theory (well, by deforming the multiplicative group associated to K-theory). His construction can be viewed as a recipe:

**start with a point of interest (an algebraic group)**

e.g., \(\mathbb{C}^\times\)

e.g., \(y^3 = x^3 – x \) over \(\mathbb{F}_3\)**deform that point (create a family of algebraic groups indexed by one parameter)**

e.g., \(\mathbb{C}^\times/q^{\mathbb{Z}}\) where \(q:= e^{2\pi i}\), we vary the norm of \(0 \leq |q| < 1\).

e.g., \(y^3 = x^3 + tx^2 – x\) over \(\mathbb{F}_3[[t]]\)- look at the formal group laws associated to your family, this is still indexed by one parameter (in fact, they can be viewed as ONE formal group law, if you keep the parameter formal)
- either apply the Landweber exact functor theorem to the whole family stalkwise (specializing the parameter), or apply the Landweber exact functor theorem to the ONE formal group law (keeping the parameter formal).

Let me say this again, because when I explain this to people they like me to say it twice.

Morava’s deformation method is a recipe which consists of 4 steps:

- construct a continuous family of smooth algebraic groups indexed by q
- construct a continuous family of formal group laws indexed by q
- construct a family indexed (indexed by q) of contra-functors from Top to AbGrp (“potential” cohomology theories — we don’t know if they are exact yet)
- prove that each member of this family of contravariant functors is exact (or treat the family as one functor, keeping the variable formal, and prove that this functor is exact)

## Derivation of the formal group law \(F_t\) and its 3 series

We take an easy example of an elliptic curve group contraction, and set it into an algebraic family, and look at how the formal group varies in the family. (This was explicitly explained to me by Dr. Lubin, thank you!)

\\(E = E_t : y^2 = x^3 + tx^2 – x\\)

The elliptic curve is defined over \(R = \mathbb{Z}[t]\), but it is not elliptic everywhere, in the sense that there are prime ideals p of Spec (R) for which E is not elliptic over \(R/p\). The discriminant of \(E_t\) is \(\Delta = 16(4-t^2)\), so we have bad reduction over the prime \(2\). To be elliptic everywhere, we invert 2 and live in the coefficient ring \(\mathbb{Z}[1/2][[t]]\).

The easiest way to get the formal group is to take the logarithm \(L(x)\) first, then find the formal group \(F\) satisfying the relation \(L(F(x,y)) = L(x) + L(y)\). We do this for our family of curves (by homogenizing, solving for \(z\), and solving for the invariant differential, \(omega\)), and find:

\begin{align*}

L(x) & \equiv x + t\frac{x^3}{3} \\

&+ (-2 + t)\frac{x^5}{5} \\

&+ (-6t + t^3)\frac{x^7}{7} \\

& + (2 – 4t^2 + \frac{1}{3}t^4)\frac{x^9}{3} \\

&+ (30t – 20t^3 + t^5)\frac{x^{11}}{11} \\

& + (-20 + 90t^2 – 30t^4 + t^6)\frac{x^{13}}{13} \mod (x^{14})

\end{align*}

Before we go on to get the formal group, notice what happens when \(3 \mid t\): then the logarithm is just \(x-2x^5/5 + 2x^9/3 – 20x^{13}/13 + …\). We get \(x^p/p\) terms at the prime for which \(y^2 = x^3 – x\) is ordinary, and no term of degree p at the primes at which the curve \(E_t\) is supersingular, but there will be a term \(x^{p^2}/p\), as we see in degree 9.

Let us now see the formal group, \(L^{-1}(L(x) + L(y))\):

\begin{align*}

F_t & \equiv x + y \\

& – t(x^2 + xy^2) \\

& + 2(x^4y + xy^4) + (4+t^2)(x^3y^2 + x^2y^3) \\

& + 2t(x^6y+xy^6) – (4t+t^3)(x^4y^3+ x^3y^4) \\

& + (-2 + 2t^2)(x^8y + xy^8) + (8+2t^2)(x^6y^3+x^3y^6) \\

& + (16 + 8t^2 + t^4)(x^5y^4 + y^4x^5) \mod (x,y)^{11}

\end{align*}

This is an expansion of the first few terms of our formal group over the polynomial ring \(\mathbb{Z}[t]\). Thus we can consider it an algebraic family of formal groups, or an analytic such. Or we can reduce it \(\mathbb{F}_p[t]\).

Doing this for p = 3, we see that if \(3 \nmid t\), we get a formal group of height one, while if \(3 | t\), the height is two, since the first 3-power terms are in degree 9. It may be clearer to write out the series:

\begin{align*}

[3]_t(x) \equiv 3x – 8tx^3 + (96 + 24t^2)x^5 + (-288t-72t^3)x^7 \\

+ (2432 + 1472t^2 + 216t^4)x^9 \mod (x^{11})

\end{align*}

What’s the moral of the story? Here, we started with an elliptic curve over (a localization of) \(\mathbb{Z}[t]\) and got a formal group over \(\mathbb{Z}[t]\), which we might reduce modulo any odd prime \(p\) to get, at that prime, a family of formal groups, not even an analytic one, but algebraic, since the coefficients were always polynomials in \(t\).

## Construction of \(\mathbb{L}^t_*(-)\)

Given a formal group \(F_t\), we may construct Lubin K-theory using Landweber-Ravenel-Stong. Let me make this clear, we are treating the formal group law family as one formal group law, otherwise, when \(3 \mid t\), our formal group law is not Landweber exact.

\\(MU^*(-) \otimes_{MU^*} \mathbb{Z}[t]\Big[\frac{1}{2}\Big] \simeq \mathbb{L}_t^*(-)\\)

Note that the \(MU^*\)-module structure induced by the genus \(g_t: MU^* \to \mathbb{Z}[t]\) associated to the formal group law \(F_t(x,y)\).

## Proof of Exactness

The exactness of each member of our family of cohomology theories \({\mathbb{L}_t | \forall t \in R}\) is shown, we examine the family in two cases. Though, really, we are treating the family as one formal group law which is exact (due to the first case)

As I mentioned before:

At \(3 \nmid t\), \(\mathbb{L}_t^*(-)\) is a Landweber-exact theory (that is, the 3-power coefficients of the 3-series of \(E_t\) is a regular sequence in \(Z[[t]]\)).

At \(3 \mid t\), \(\mathbb{L}_t^*\) is not Landweber-exact, but is constructible via EKMM by directly killing elements in \(MU\). (This is very similar to the construction for Morava K-theory localized at the prime 3, \(K(2)_{(3)}\))

Let’s prove this.

Indeed, that it is not Landweber-exact suggests that Lubin K-theory, at \(3 | t\), behaves like Morava K-theory of height 2; and in general, that Lubin K-theory behaves like a variant of a Morava E-theory of height 2 at the prime 3 (it is in fact Bousfield equivalent).

**Note on a closely related family:** The family independently constructed by Behrens, \\(y^2 = 4x^3 + u_1x^2 – 2x\\) as a universal deformation of the curve \(y^2 = x^3 – x\), the deformation \\(y^2 = x^3 + tx^2 – x\\) is probably also a universal deformation, which would make \(\mathbb{L}_t\) a model for Morava \(E_2\) at the prime \(3\).

We’ve shown that \(\mathbb{L}^*_t\) is a cohomology theory by directly applying the Landweber exact functor theorem. QED.

Now that we’ve constructed our cohomology theory \(\mathbb{L}_t\), let’s try it out on some routine spaces…

## Some computations

Claim: \\(E^*(B\mathbb{Z}/p) = E^*(\mathbb{C}P^\infty)/[p]_F(x)\\) for any complex orientable cohomology theory \(E\), with associated formal group \(F\), where \([p]_F(x)\) denotes the p-series of its associated formal group law \(F\).

- There is an exact sequence of groups \\(Z \xrightarrow{p} \mathbb{Z} \to \mathbb{Z}/p\\)
- Applying “H” gives a cofiber sequence of spectra \(H\mathbb{Z} \xrightarrow{p} H\mathbb{Z} \xrightarrow H\mathbb{Z}/p\). It extends to the right to give the iterated cofiber sequence \\(H\mathbb{Z} \xrightarrow{p} H\mathbb{Z} \to H\mathbb{Z}/p \to \Sigma H\mathbb{Z} \xrightarrow{p} \Sigma H\mathbb{Z} \to \Sigma H\mathbb{Z}/p \to \Sigma^2 H\mathbb{Z} \xrightarrow{p} \Sigma^2 H\mathbb{Z} \to …\\)
- For spectra, cofiber sequences are the same as fiber sequences, so the above is also an iterated fiber sequence.
- Taking \(\Omega^\infty\) preserves fiber sequences, so the last few nodes give a fiber sequence \(K(Z, 1) \to K(Z/p, 1) \to K(Z, 2) \xrightarrow{p} K(Z, 2)\) of spaces.
- The data of this iterated fiber sequence is equal to the data of the pullback square of fiber sequences

Note that, \(K(Z, 1) = S^1, K(Z/2, 1) = RP^\infty,\) and \(K(Z, 2) = CP^\infty\). As an example, in the case where \(p = 2\): \\(E^*(\mathbb{R} P^\infty) = E^*(\mathbb{C} P^\infty)/[2]_F(X)\\):

Returning to the general case, using the AHSS, \\(E^{p,q}_2 = H^p_{cell} (\mathbb{C} P^\infty; E_qS^1) \Rightarrow E^{p+q}(RP^\infty)\\)

Theorem (Gysin): For spherical fibrations, there is *one* differential, multiplication by the euler class (\(d_2(e) = e(\mathcal{L}\))).

From this, we know that \(e(p^*\mathcal{L}) = p^*(e(\mathcal{L})) = p^*(x) = [p]_F(x)\).

Examples: \(\mathbb{K}^*(B\mathbb{Z}/2); \mathbb{L} ^t_*(B \mathbb{Z}/2); \mathbb{L}^t_*(B\mathbb{Z}/3)\).

We see that it collapses to \(K^*(RP^\infty) = \mathbb{Z}[\beta^{pm}][[x]]/(2x + x^2)\).

In the case of Lubin K-theory, the 2-series is a unit in \(\mathbb{L}^*(CP^\infty)\), so \\(\mathbb{L}^*(RP^\infty) = 0\\) This is reasonable, for the homotopy groups of \(RP^\infty\) are trivial when tensored with \(Z/3\), and \(\mathbb{L}_t\) is inherently \(3\) local due to our family of elliptic curves living over the ring \(\mathbb{F}_3\).

I will state this again to emphasize it’s role: We see that \([p]_F(x)\) is a unit in \(E^*(CP^\infty)\) if \(E\) is q-local and \(p \neq q\). Thus, if \(p \neq 2\), \(E^*(RP^\infty) = *\).

We see that it collapses to \(\mathbb{L}^*(BZ/3) = \mathbb{Z}[\beta^{pm}][[x]]/[3]_{t}(x)\).

**Sidenote:** This is all assuming that we have an even periodic family, that is, we started with an ungraded formal group law \(F(x,y)\), and graded it with our periodic element u by *multiplicatively* conjugating \(u^{-1}F(ux, uy)\) (we are NOT *compositionally* conjugating by an invertible power series \(u\), which would be \(u^{-1}(F(u(x), u(y))\)). We’ve started in degree 0, and added \(u\) to let us hop up and down degrees. **Open question:** But what if \(F\), our formal group law, is already graded? I’m not sure!

I can not help but wonder if this method generalizes to the creation of other (besides, say, \(x + y + txy\)). cohomology theories which exhibit “height contraction” behaviour. The next hope is for a construction of height 3 which contracts to height 2. (This remains a hope because I am having difficulty writing down a simple example of a K3-variety which has a one-parameter contraction to a supersingular elliptic curve.)

*Sidenote: we use “deformation” in the general setting of viewing any deformation as the replacement of a point \(\text{Spec }k\) with a fat point \(\text{Spf }k[[t]]\).*

In summary, at risk of using “pyrotechnical” language:

Morava deforms K-theory by deforming the underlying formal group law, \(\mathbb{G}_m\), to the height 1 elliptic curve group \(\mathbb{G}_m/q^\mathbb{Z}\); in modern terms, he constructs tmf in a neighborhood of the cuspidal point of \(M_{\ell}^+\) (over \(\mathbb{Z}\)). This is now called “Tate K-theory.”

We constructed a cohomology theory over a neighborhood of a supersingular point \(C\) of \(M_{\ell}\) by deforming the underlying height 2 elliptic curve group, \(C\), into a height 1 elliptic curve group. We call this “Lubin K-theory,” it’s not all Landweber exact, so we didn’t use the same proof method as Morava’s beautiful equivariant sheaf argument.

## Some Absurd Speculations: “Chromatic” Coppersmith theory

Don Coppersmith, a student of Sternberg, constructed an answer to the question:

What is a theory of the behavior of homogeneous symplectic manifolds as their parent Lie groups undergo contraction?

This question alone is a beautiful perspective, and suggests a slight adjustment of the traditional approach to chromatic homotopy theory.

What is a theory of the behavior of complex-orientable cohomology theories as their parent smooth algebraic groups undergo contraction?

I’m particularly curious how to relate this theorem by (Chu):

To every simply connected Lie group \(G\) with Lie algebra \(\mathfrak{g}\), every left-invariant closed 2-form induces a symplectic homogenous space.

There is a bijective correspondence between the orbit spaces of 2-cocycles of a given Lie algebra, and equivalence classes of simply-connected symplectic homogenous spaces of the Lie group.

To this quote of Morava (and maybe also to the symmetric 2-cocycles of Lazard):

“Height is a complete invariant for (one-dimensional) formal groups over a separably closed field; in other words, in characteristic p the set of geometric points of \(\Lambda\) stratifies into orbits indexed by the set \(\mathbb{N} \cup \infty\). The orbit of a grouplaw \(F\) is a homogeneous space \(\Gamma/S_F\) and an equivariant sheaf of modules over such a quotient is locally free; indeed, it can be recovered from the action of the isotropy group \(\Gamma/S_F\) of the orbit on the fiber of the sheaf above the point \(F\).”

Why bother to translate Coppersmith theory into the language of formal group laws?

The deformation theory of Lie algebras seems to be pretty hands-on and well developed, and I am a bit desperate for geometric intuition/computational help in thinking about height 2 and height 3 cohomology theories. For example: it would be pretty cool if we could literally deform the model of topological K-theory in complex line bundles to something like bundles with a Poisson-Lie group structure on them (giving us an associative bracket thing a la Poisson geometry). This probably doesn’t make sense, but something like it might!

It’s open how much of Coppersmith’s theory generalizes to our case. In other words, I’m working on it and don’t understand the machinery yet — however, there is hope that the machinery can be translated because both settings can be rephrased as theorems about Hopf Algebras [probably]. [A section in Coppersmith’s thesis that seems to relate directly to our story is \(t-p\)-contractions, or “even-odd”-contractions) which tell us something about the existence of stable isotopy subalgebras.]

## A Quick Note on a Geometric Definition of \(v_n\)

*This post assumes knowledge of the definition of the oriented cobordism ring, as well as the equivalence \(\pi_*MU \simeq MU^*(pt) =: MU^*\), and familiarity with the Landweber exact-functor theorem.*

A quick post on a nice thing. I was reading Quillen and stumbled across what seems to be the first nod toward the importance of the coefficients \(p, v_1, v_2, … \in MU^*\).

I have complained about my confusion wrt these coefficients and the concept of complex orientation in a few past blog posts. I’ve read about it so many times in many different equivalent forms, but finally, this one stuck.

They are defined as normal bundles which correspond to a choice of weakly complex structure. Thanks to Tyler Lawson for confirming and clearing up my suspicions on this connection.

For those who haven’t encountered homotopy theory, I’d like to show you the following excerpt of Whitehead so you may realize that Quillen is observing a creature in its native habitat.

“A complex orientation of a map of manifolds \(f: Z \to X\) is a generalization of a weakly complex structure on \(Z\) when \(X\) is a point. By a complex orientation of \(f\), we mean an equivalence class of factorizations of \(f\),

\\(Z \xrightarrow{i} E \xrightarrow{\pi} X\\)

where \(p: E \to X\) is a complex vector bundle and \(i\) is an embedding endowed with a complex structure on its normal bundle \(v_i\).”

Does the normal bundle \(v_i\) have to do with the \(v_i\) in the sequence \((p, v_1, v_2, …)\) in the coefficient ring of MU? Are these just the same letters being used?

Since an equivalence class of factorizations of \(f\) is an equivalence of choices of the embedding \(i\), then this is also an equivalence class of the complex structures of the normal bundles \(v_i\).

Next, if we take the equivalence class of factorizations Z -i-> E -p-> X to be up to cobordism, then each \(v_i\) is represented by an element of \(MU^*(X)\). We have to choose \(X\) to be \(pt\) for the \(v_i\) to be in the coefficient ring as we’d expect.

*Sidenote: Quillen also mentions that if the dimension of E is “sufficiently large” then one obtains each complex orientation of \(f\) from exactly one homotopy class of complex structures on \(v_i\). I am not sure why \(E\) must be large for this to be true, but I have been told it is for the following two reasons. One is that \(E\) has to be sufficiently large for \(Z\) to be able to embed. The second is that it also has to be large for some ambiguities in the process (the isomorphism class of the normal bundle, for example) to be eliminated.*

*Another sidenote: here’s the excerpt from Whitehead which is not as directly relevant:*

It seems that Dan Quillen’s guiding conviction was to understand a mathematical phenomena by seeking out its very simplest concrete manifestation. Due to this, I doubly appreciated the following quote from this seminal paper of his that we’ve been talking about:

I have been strongly influenced by Grothendieck’s theory of motives and like to think of a cobordism theory as a universal contravariant functor on the category of \(C^\infty\) manifolds endowed with Gysin homomorphism for a class of proper “oriented” maps, instead of as the generalized cohomology theory given by a specific Thom spectrum.

—-Dan Quillen, [Elementary Proofs of Results in Cobordism Theory]

He introduced formal groups as a tool in algebraic topology due to his interest to understand from first principles the result from cobordism theory that the coefficient ring of \(MU\) is a polynomial ring with an infinite number of even generators.

This caught on. If you’re curious wrt these \(v_n\), I wrote this little Toda-Smith article on nlab which brushes by how they come up as periodic maps.

An awareness of the classification results on formal group laws gives us some computational tools to try and wrassle the impossible beast of the homotopy groups of spheres. We usually do this a prime at a time, that is, study the homotopy groups of the p-local sphere, because it is way easier. Still impossibly hard, but easier.

For example, his methods led to our finding that \(v_{n}^{-1} BP_*/(p^\infty, …, v_n^\infty\) is like \(H^*_{gp}(\mathbb{S}_n, E_*)\). People (who actually know how to ram their heads against these things!) compute some of the higher ones (?) by playing a super-hard game along these lines:

- \(Ext_{BP_*BP}^{*,*}(v_{n}^{-1} BP_*/(p^\infty, …, v_n^\infty)\)
- apply the chromatic spectral sequence to get \(Ext^{*,*}BP_*\) ,
- then apply the Adams-Novikov to get \(\pi_*\mathbb{S}_{(p)}\)

## Basic Topology

#### Topological Maps in Robotic Navigation

The goal for an autonomous robot is to be able to construct (or use) a map or floor plan and to localize itself in it. The discipline of robotic mapping is concerned with map *representation*.

The internal representation of a map can be “metric” or “topological”:

- A
**metric framework**(the most common for humans) considers a two-dimensional space in which it places the objects. The objects are placed with precise coordinates. This representation is very useful but is sensitive to noise and it is difficult to calculate the distances precisely. - A
**topological framework**only considers places and relations between them.- This topological map is a
*graph-based representation of the environment*where certain easily distinguishable places in the environment, labeled as landmarks, are designed as*nodes*. - The
*edges*are deemed to represent navigable connections. In addition, the edges of the topological graph may be annotated with information relating to navigating the corresponding regions in the environment. This framework is more resistant to noise and generally easier to store internally.

- This topological map is a

**The robot needs to know what room it’s in and what doors it needs to pass through to get to the required location.** *This does not require the dimensions and shape of the rooms.*

This post highlights how the axioms governing the metric space naturally lead into its abstraction: the topological space.

#### The Metric Space

A metric space is:

- a set of \(X\)
- a distance function \(d(x,y)\) which defines the distances between all members of the set.

For example, the distance function on the real line \(\mathbb{R}^1\) is \(d(x,y) = |x-y|\).

This distance function takes a pair of 2 ordered numbers \((x,y)\) and gives us another real number (the distance between \(x\) and \(y\)). We can easily generalize this distance function from the real line \(\mathbb{R}\) to any n-dimensional space \(\mathbb{R}^n\):

\(d: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n\)

We can have two different distance functions on the same space \(\mathbb{R}^2\)

For each pair of points \((x_0, y_0)\) and \((x_1, y_1)\) \(\mathbb{R}^2\),

the normal distance function is derived from the Pythagorean therorem:

- \(d ((x_0, y_0), (x_1, y_1)) = \sqrt{(x_0 – x_1)^2 + (y_0-y_1)^2}\)

With this distance function, \(d(x, a) \leq 1\) for a fixed point \(a \in \mathbb{R}^2\).

- \(d'((x_0, y_0), (x_1, y_1)) =\) maximum \({|x_0-x_1|, |y_0 – y_1|}\).

The distance function of a Hilbert space is

- \(d(u,v) = [\sum\limits_{i=1}^{\infty} (u_i – v_i)]^{1/2}\)

#### Neighboorhoods

The concept of a neighborhood is powerful, for it formalizes the concept at the heart of analysis: continuity.

An open neighborhood of a point \(p\) in a metric space \((X, d)\) is the set \(N_\epsilon = {x \in X | d(x, p) < \epsilon}\).

Example:

- If \(p \in \mathbb{R}\) then the open-interval of \(p\) is \((p – \epsilon, p + \epsilon)\) of radius \(\epsilon\) centered at \(a\).
- The open disc and open ball, which are the 2D and 3D analogues of the open interval. (I haven’t explained these yet, I’ll formally cover this example of a neighborhood later)

A function \(f\) is continuous over a space \(X\) iff \(\forall x, p \in X\) the following condition is satisfied:

\(\forall \epsilon \exists \delta\) \(|\) \(x \in N_\epsilon(p) \rightarrow f(x) \in N_\delta(f(p))\)

This formal understanding of continuity leads directly to the unifying concept of this lecture, the study of “closeness”: the open set.

A subset \(S\) of a metric space \(X\) is an open set iff every point of \(S\) has an open neighbourhood \(N_\epsilon\) that lies completely in \(S\). In precise terms:

\(S\) is open \(\Leftrightarrow \forall a \in S \exists \epsilon > 0 | N_\epsilon(a) \subseteq S\).

#### Properties of Open Sets

1. The union (of an arbitrary number) of open sets is open.

**Proof**

Let \(x \in \bigcup\limits_{i \in I} A_i = A\). Then \(x \in A_i\) for some \(i\). Since \(A_i\) is open, by the definition of an open set: \(x\) has an open neighbourhood lying completely inside \(A_i\) which is also inside \(A\). \(_{QED}\)

**Proof**

To check that finite intersections of open sets are again open, it suffices to check this for the intersection of two open sets. Suppose \(x \in A \cap B\).

Then \(x \in A\) and also has a neighbourhood, \(N_{\epsilon_A}\) lying in \(A\). Similarly, \(x\) has a neighbourhood \(N_{\epsilon_B}\) lying in \(B\). If \(\epsilon =\) min \({\epsilon_A, \epsilon_B}\), the neighbourhood \(N_\epsilon\) also lies in both \(A\) and \(B\) and hence in \(A \cap B\). \(_{QED}\)

#### Continuous Functon

In analysis, we examine the convergence of sequences, the continuity of functions, and the compactness of sets. Some types of convergence, such as the pointwise convergence of real-valued functions defined on an interval, cannot by expressed in terms of a metric on a function space.

Topological spaces provide a general framework for the study of convergence, continuity, and compactness. The fundamental structure on a topological space is not a distance function, but a collection of open sets.

A continuous function can be defined in terms of open sets.

If \(f: X \rightarrow Y\) is a continuous function between metric spaces and \(B \subset Y\) is open, then \(f^{-1}(B)\) is an open subset of \(X\).

**Proof**

Let \(x \in f^{-1}(B)\). Then \(f(x) = y in B\).

Since \(B\) is open, the point \(y\) has a neighbourhood \(N_\epsilon \subset B\).

By the definition of continuity, \(N_\epsilon\) contains the image of some neighbourhood \(N_\delta\) \(V\) of \(x\). Since \(f(V) \subset B\), we have \(V \subset f^{-1}(B)\) and so \(x\) has this nieghbourhood \(N_\delta \subset f^{-1}(B)\).

Hence, the definition of an open set, \(f^{-1}(B)\) is open in \(X\).

The converse also holds: If \(f: X \rightarrow Y\) is a function for which \(f^{-1}(B)\) is open for every open set \(B\) in \(Y\). Proving this here would only take the joy of proving this away from the reader: it’s a short proof, you can do it!

#### The Open Ball: Interior and Boundary Points

As we mentioned earlier: in \(\mathbb{R}^1\), the open neighbourhood is the open interval. In \(\mathbb{R}^2\) it is the open disc. In \(\mathbb{R}^3\) it is the open ball.

An \(n\)-dimensional open ball of radius \(r\) is the collection of points of a distance less than \(r\) from a fixed point in Euclidean \(n\)-space. Explicitly, the open ball with center \(p\) and radius \(r\) is defined by:

\(B_r (p)= {x \in \mathbb{R}^n | d(p,x) < r}\)

if \(X\) is a metric space with metric \(d\), then \(x\) is an interior point of \(S\) if there exists \(r > 0\), such that \(y\) is in \(S\) whenever the distance \(d(x,y) < r\).

Equivalently, if \(S\) is a subset of a metric space, then \(x\) is an interior point of \(S\) iff (there exists an open ball with a radius larger than 0 centered at \(x\) which is contained in \(S\)):

\(\exists r\) \(|\) \(x \subseteq B_r (x)\)

A friend of the interior point is the definition we’ve been dancing around: a boundary point.

Intuitively, a point is an interior point of it is not “right on the edge” of a set, and a boundary point if it is “right on the edge” of a set.

Formally, a point \(p\) is a boundary point of \(S\) iff for every \(\epsilon > 0\), the open neighbourhood of \(p\) intersects both \(S\) and the complement of \(S\): \(\bar{S}\). That is: \(x\) is a boundary point of \(S\) iff:

\(N_\epsilon(p) \cap S \neq \emptyset\) and \(N_\epsilon(p) \cap \bar{S} \neq \emptyset\)

Sidenote: The complement \(\bar{S}\) of \(S\) is \(U\) \(S\), the set of all things outside of \(S\). This can be rephrased as all things (in the universal set \(U\)) that are not in \(S\).

#### Topological Balls

We may talk about balls in any space \(X\), not necessarily induced by a metric.

An (open or closed) \(n\)-dimensional topological ball of \(X\) is any subset of \(X\) which is homeomorphic to an (open or closed) Euclidean \(n\)-ball. Topological \(n\)-balls are important in combinatorial topology, as the building blocks of cell complexes.

#### Toplogical Spaces

Recall that a metric space is just a set \(S\) with a metric defined on it.

A topological space is a set \(S\) with a geometric structure defined on it.

Abstractions of the properties of open sets are the axioms that define a topology \(T\). Moreover, we call an element of \(T\) is an open set.

1. The union (of an arbitrary number) of open sets is open.

=> The union of any set of members of \(T\) is in \(T\).

2. The intersection of finitely many sets is open.

=> The intersection of finitely many members of \(T\) is in \(T\).

For completeness, the whole set \(S\) and \(\emptyset\) are also in \(T\).

Notational aside: We call the pair \((S, T)\) a topological space; if \(T\) is clear from the context, then we often refer to \(S\) as a topological space.

3. If \(f: X \rightarrow Y\) is a continuous map between topological spaces and \(B \subset Y\) is open, then \(f^{-1}(B)\) is an open subset of \(X\).

#### Basis of a Topology

Let’s construct a topology of the reals.

We want to know the minimum data you need to specify a structure defined on a set. In many cases, this minimum data is called a basis and we say that the basis generates the structure.

The collection of open intervals of the form \((p – \epsilon, p + \epsilon)\) on \(\mathbb{R}\) is not a topology. The collection of such intervals doesn’t include the empty set, the whole line, or the union \((1,2) \cup (3,4)\).

The usual topology is not defined as the collection of such intervals. Instead, the topology is defined as the collection of all possible unions of such intervals. In other words, the intervals of the form \((p – \epsilon, p + \epsilon)\) are a basis for the topology. It’s important to not confuse the basis with the whole topology.

The metric topology is the topology on a set \(X\) generated by the basis \({B_\epsilon(x) | \epsilon > 0, x \in X}\).

#### In many areas of mathematics we would like to endow our sets with additional structures, such as a metric, a topology, a group structure, and so forth.

It is here that I will stop, this post, but I will give you some keywords to look up:

Linear Functionals and Bilinear Forms, Riesz Representation Theorem, Linear, Bounded, Continuous, comparing topologies: \((S, T’)\) has more open subset to separate 2 points in \(S\) than \((S, T)\).

Topology of Robot Motion Planning

Topological Robotics: Motion Planning in Projective Spaces

http://www.math.ucla.edu/~tao/preprints/compactness.pdf

Basic Facts About Hilbert Space

Consistent Observation Grouping for Generating Metric-Topological Maps that Improves Robot Localization

Mobile Robots Navigation, Mapping, & Localization: Part II

Alegbraic Topology for Computer Vision