These notes are based on the Fall 2024 lecture notes by Stephen New at the University of Waterloo, substantially enriched with motivation, examples, and narrative to serve as a self-contained introduction to point-set and elementary algebraic topology. The reader is assumed to have experience with real analysis (metric spaces, continuity, compactness) and basic group theory.

Part I: Point-Set Topology

Chapter 1: Topological Spaces and Continuous Maps

The leap from metric spaces to topological spaces

The story of topology begins with a simple but powerful observation about metric spaces. In a course on real analysis, one learns that a subset \(U\) of a metric space \((X, d)\) is open if every point \(a \in U\) has some elbow room — there exists \(r > 0\) such that the open ball \(B(a, r) = \{x \in X \mid d(a,x) < r\}\) is entirely contained in \(U\). One then proves three fundamental properties of the collection of all open sets:

1. The empty set \(\emptyset\) and the whole space \(X\) are both open.
2. The union of any collection of open sets is open.
3. The intersection of any finite collection of open sets is open.

Why only finite intersections? Consider the open intervals \((-1/n, 1/n)\) in \(\mathbb{R}\). Each is open, but their intersection \(\bigcap_{n=1}^{\infty} (-1/n, 1/n) = \{0\}\) is a single point — decidedly not open. Finiteness is essential.

The leap of abstraction that defines topology as a discipline is this: what if we forgot about the metric and kept only these three properties? If we declare that a “topology” on a set \(X\) is any collection of subsets satisfying these axioms, we obtain a vastly more general framework that captures continuity, convergence, and connectedness without reference to distances. This is the concept of a topological space.

Topological spaces

Note that to verify property (3), it suffices to check pairwise intersections: if \(A, B \in \mathcal{T}\) implies \(A \cap B \in \mathcal{T}\), then induction handles any finite collection.

When \(a \in X\), an open neighbourhood of \(a\) is any open set \(U \in \mathcal{T}\) with \(a \in U\). The notion of “being near a point” is entirely determined by which open sets contain it.

When two topologies \(\mathcal{S}\) and \(\mathcal{T}\) on the same set satisfy \(\mathcal{S} \subseteq \mathcal{T}\), we say \(\mathcal{S}\) is coarser (fewer open sets, harder to separate points) and \(\mathcal{T}\) is finer (more open sets, more discriminating). This terminology reflects a lattice of topologies ordered by inclusion.

First examples

Every metric space gives a topology (the metric topology), but there are two extreme cases that require no metric at all.

Closed sets, interior, closure, and boundary

The interplay between open and closed sets is one of the central themes of point-set topology. Closed sets are precisely the complements of open sets, and they satisfy dual axioms.

A set can be both open and closed (such as \(\emptyset\) and \(X\) in any topology, or every set in the discrete topology), or neither (such as \([0,1)\) in \(\mathbb{R}\)).

The interior and closure are idempotent operations: \((A^\circ)^\circ = A^\circ\) and \(\overline{\overline{A}} = \overline{A}\). A set is open if and only if \(A = A^\circ\), and closed if and only if \(A = \overline{A}\). Every set decomposes as the disjoint union \(\overline{A} = A^\circ \cup \partial A\).

There is a useful pointwise characterization of these concepts. A point \(a\) lies in \(A^\circ\) precisely when some open neighbourhood of \(a\) is entirely contained in \(A\). A point \(a\) lies in \(\overline{A}\) precisely when every open neighbourhood of \(a\) meets \(A\).

Bases for topologies

Specifying every open set in a topology can be unwieldy. A basis provides a compact description from which the entire topology can be recovered, much like how a basis for a vector space generates the whole space.

A useful criterion: if \(\mathcal{B} \subseteq \mathcal{T}\), then \(\mathcal{B}\) is a basis for \(\mathcal{T}\) if and only if for every \(a \in X\) and \(U \in \mathcal{T}\) with \(a \in U\), there exists \(B \in \mathcal{B}\) with \(a \in B \subseteq U\). In other words, every open set can be “approximated from inside” by basis elements.

A powerful structural result underlies the flexibility of topological spaces: any collection \(\mathcal{S}\) of subsets of \(X\) generates a unique smallest topology containing \(\mathcal{S}\), because the intersection of any collection of topologies is again a topology.

Hausdorff spaces

The axioms of a topological space are deliberately weak — so weak that they allow pathological behaviour. In the trivial topology, for instance, no two distinct points can be separated by open sets. To exclude such degeneracies, we impose separation axioms.

The Hausdorff condition ensures that limits of sequences (when they exist) are unique, and that one-point sets are closed. Every metric space is Hausdorff: given \(a \neq b\), take \(r = \frac{1}{2}d(a,b)\) and use the balls \(B(a,r)\) and \(B(b,r)\).

The Hausdorff property is so natural that many authors include it in the definition of a topological space. We do not, but the reader should be aware that nearly all spaces encountered in analysis, geometry, and algebra are Hausdorff.

The subspace topology

Given a topological space \(Y\) and a subset \(X \subseteq Y\), there is a natural way to make \(X\) into a topological space.

This is indeed a topology on \(X\) (the verification is straightforward). A subset \(A \subseteq X\) is closed in the subspace topology if and only if \(A = B \cap X\) for some closed \(B\) in \(Y\). When a metric space \(Y\) has a subset \(X\), the subspace topology coincides with the metric topology on \(X\) using the restricted metric.

Continuous maps and homeomorphisms

The concept of continuity generalizes naturally from metric spaces: instead of the \(\varepsilon\)-\(\delta\) definition, we use the “inverse image of open sets is open” formulation.

Homeomorphism is the fundamental equivalence relation in topology. Two homeomorphic spaces are, from the topological viewpoint, identical — they have the same open sets, the same convergent sequences, the same connected components, and so on. The central problem of topology is to classify spaces up to homeomorphism and to find invariants that distinguish non-homeomorphic spaces.

A useful technique for verifying continuity: if \(\mathcal{C}\) is a basis for the topology on \(Y\), then \(f: X \to Y\) is continuous if and only if \(f^{-1}(C)\) is open in \(X\) for every \(C \in \mathcal{C}\). This often simplifies checking continuity enormously.

Continuity can also be verified locally: if \(X\) is covered by open sets \(\{A_k\}\) and each restriction \(f|_{A_k}: A_k \to Y\) is continuous, then \(f\) is continuous. A similar “gluing lemma” holds for finitely many closed sets.

Chapter 2: Examples of Topological Spaces

The standard topology on Euclidean space

The most important topology in all of mathematics is the standard topology on \(\mathbb{R}^n\), induced by the standard metric \(d(x,y) = \sqrt{\sum_{k=1}^n (y_k - x_k)^2}\). Unless stated otherwise, subsets of \(\mathbb{R}^n\) always carry this topology (which coincides with the subspace topology).

The order topology

Metric spaces are not the only natural source of topologies. Any linearly ordered set carries a natural topology defined by intervals.

The standard topology on \(\mathbb{R}\) is the order topology for the usual ordering. But the construction applies to any ordered set — the ordinals, the long line, the dictionary-ordered plane — yielding many interesting spaces.

A variation is the lower limit topology on \(\mathbb{R}\), denoted \(\mathbb{R}_\ell\), whose basis consists of half-open intervals \([a, b)\). This is strictly finer than the standard topology (every open interval \((a,b)\) is a union of sets \([c, b)\) with \(c > a\), but \([a, b)\) is not open in the standard topology). The lower limit topology, also called the Sorgenfrey line, provides many counterexamples in general topology.

Product spaces

Given two topological spaces \(X\) and \(Y\), we want to put a natural topology on the Cartesian product \(X \times Y\). The right choice is dictated by the requirement that the projection maps should be continuous.

This universal property is the real reason for the product topology — it is the unique topology making the product into a categorical product in the category of topological spaces.

Infinite products and the box vs. product topology

For an indexed family of spaces \(\{X_k\}_{k \in K}\), there are two natural candidates for a topology on the Cartesian product \(\prod_{k \in K} X_k\).

For finite index sets, the two topologies coincide. For infinite index sets, the product topology is strictly coarser. Why prefer the product topology? Because it satisfies the universal property: a map \(f: Z \to \prod X_k\) is continuous (in the product topology) if and only if each component \(p_k \circ f: Z \to X_k\) is continuous. The box topology breaks this — continuity of each component does not imply continuity of the product map. The product topology is also the one that makes Tychonoff’s Theorem (compactness of products) work.

The quotient topology

Quotient spaces are one of the most important constructions in topology, allowing us to “glue” points together and build new spaces from old ones.

The quotient topology is the finest topology making the quotient map continuous. It has a useful universal property: a map \(f: X/{\sim} \to Y\) is continuous if and only if \(f \circ q: X \to Y\) is continuous.

A particularly clean source of quotient spaces comes from group actions.

Not all quotient constructions yield nice spaces. The action of \((0, \infty)\) on \(\mathbb{R}^2\) by scaling gives a quotient space that is not Hausdorff — the equivalence class of the origin cannot be separated from any ray.

Chapter 3: Connected, Path-Connected, and Compact Spaces

These three properties — connectedness, path-connectedness, and compactness — are the workhorses of topology. They are topological invariants: preserved under homeomorphism. This makes them powerful tools for proving that two spaces are not homeomorphic.

Connectedness

Intuitively, a space is connected if it is “in one piece.” The formal definition captures this by saying the space cannot be partitioned into two non-trivial open sets.

This is the key to many applications: if \(f: X \to Y\) is continuous and \(X\) is connected, then \(f(X)\) is connected. In particular, homeomorphic spaces share the same connectedness properties.

This characterization is equivalent to the completeness of the real numbers (the least upper bound property).

This “chain lemma” is the tool for building large connected sets from small ones: as long as the pieces overlap, their union stays connected.

The proof for arbitrary products is more delicate — one shows that certain finite-dimensional “slices” are connected (by the finite case), their union is connected (they share a common point), and this union is dense.

Connected components

Every topological space decomposes uniquely into its maximal connected pieces.

Each connected component is connected and closed (being the closure of a connected set, which is connected). The connected components of \(\mathbb{Q}\) are the singletons — the rationals are totally disconnected.

Path-connectedness

There is a stronger and more geometric notion of connectedness: the ability to walk continuously between any two points.

The converse is false. The topologist’s sine curve \(\overline{A}\), where \(A = \{(x, \sin(1/x)) \mid x > 0\}\), is connected but not path-connected: there is no path from \(A\) to the segment \(\{0\} \times [-1, 1]\) despite the closure connecting them.

Every convex subset of a normed space is path-connected (the straight line between any two points lies in the set), providing a rich supply of path-connected spaces.

Path-components are defined analogously to connected components, using the equivalence relation of being connected by a path. Each path-component is contained in a single connected component.

Compactness

Compactness is perhaps the most important topological property. It is a topological generalization of “finiteness” — a compact space, while potentially infinite, behaves in many ways like a finite set.

Why is compactness important? It allows us to pass from local information to global conclusions. If something is true “near every point” (captured by an open cover), compactness guarantees it is true “uniformly” (captured by finitely many sets).

This immediately implies the extreme value theorem: a continuous real-valued function on a compact space achieves its maximum and minimum.

The second statement requires the Hausdorff condition — in a non-Hausdorff space, compact subsets need not be closed. The proof constructs, for each point outside the compact set, a pair of disjoint open sets using the Hausdorff property, then extracts a finite subcover to build a single open set separating the point from the compact set.

This remarkable result says that a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism — the inverse is “free.” This is used constantly in topology to avoid the labour of proving the inverse is continuous.

Tychonoff’s Theorem

The crown jewel of point-set topology is:

The proof uses Zorn’s Lemma (equivalently, the Axiom of Choice) and the finite intersection property: a space is compact if and only if every collection of closed sets with the finite intersection property (every finite subcollection has non-empty intersection) has non-empty total intersection. This equivalence is the “closed set” dual of the open cover definition, obtained by taking complements.

Tychonoff’s Theorem is powerful because it applies to arbitrary products. It underpins much of functional analysis (the Banach-Alaoglu theorem on weak-\(*\) compactness relies on it) and algebraic geometry.

Part II: Separation, Countability, and Manifolds

Chapter 4: Countability and Separation Axioms

Countability axioms

How “large” is a topology? The countability axioms measure this using the lens of countability.

These properties are ordered by strength: second-countable implies all the others. In metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.

Separation axioms

The separation axioms form a hierarchy measuring how well the topology distinguishes points from closed sets.

The hierarchy is: normal \(\Rightarrow\) regular \(\Rightarrow\) Hausdorff \(\Rightarrow\) \(T_1\). Every metrizable space is normal. Each implication is strict — there exist spaces at each level that fail the next.

Regularity has a useful equivalent formulation: \(X\) is regular if and only if for every point \(a\) and open \(W\) with \(a \in W\), there exists open \(U\) with \(a \in U \subseteq \overline{U} \subseteq W\). This says we can always “shrink” an open neighbourhood to get a closed neighbourhood.

Urysohn’s Lemma and its consequences

The separation axioms gain their power through Urysohn’s Lemma, one of the deepest results in point-set topology.

The proof is a beautiful construction: one builds a family of open sets \(\{U_r\}_{r \in \mathbb{Q} \cap [0,1]}\) with \(\overline{U_r} \subseteq U_s\) whenever \(r < s\), and defines \(f(x) = \inf\{r \in \mathbb{Q} \mid x \in U_r\}\). The nested family of sets is built by induction along an enumeration of the rationals.

Urysohn’s Lemma leads to two major theorems:

The proof of the Tietze Extension Theorem builds the extension as a uniformly convergent series, using Urysohn’s Lemma at each step to approximate the function more closely.

Chapter 5: Topological Manifolds

Manifolds are the spaces where geometry and topology meet. They are the natural setting for calculus on curved spaces, and their study lies at the heart of modern mathematics and physics.

Definition and examples

The Hausdorff condition and second-countability exclude pathological examples (like the “line with two origins”) while still being general enough to include all spaces of geometric interest.

When two charts \(\varphi, \psi\) overlap, the composition \(\psi \circ \varphi^{-1}\) (a transition function) is a homeomorphism between open subsets of \(\mathbb{R}^n\). By imposing smoothness on these transition functions, one obtains a smooth manifold — the setting for differential geometry.

Projective spaces

Projective space is an \(n\)-manifold: it is covered by \(n + 1\) charts \(U_k = \{[x_1, \ldots, x_{n+1}] \mid x_k \neq 0\}\), each homeomorphic to \(\mathbb{R}^n\) via \(\varphi_k([x]) = (x_1/x_k, \ldots, \widehat{x_k/x_k}, \ldots, x_{n+1}/x_k)\). Every manifold is regular, hence metrizable by the Urysohn Metrization Theorem.

One can also show \(\mathbb{P}^n \cong S^n / \{\pm 1\}\), the sphere with antipodal points identified.

Classification of compact surfaces

One of the triumphs of 19th and early 20th century topology is the complete classification of compact connected 2-manifolds (surfaces) up to homeomorphism.

Here the connected sum \(X \# Y\) is formed by removing a small open disc from each surface and gluing along the resulting boundary circles. The sphere \(S^2\) is the identity: \(X \# S^2 \cong X\).

The proof proceeds through several stages:

1. Triangulability: Every compact 2-manifold can be triangulated, hence represented as a polygon with edges identified in pairs.
2. Cut-and-paste operations: Systematic moves (cutting along diagonals, reattaching along identified edges) reduce any edge-identification word to a standard form.
3. Standard forms: Orientable surfaces give the word \(a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1}\); non-orientable surfaces give \(a_1 a_1 a_2 a_2 \cdots a_h a_h\).
4. Key relation: \(\mathbb{P}^2 \# T^2 \cong \mathbb{P}^2 \# \mathbb{P}^2 \# \mathbb{P}^2\) (attaching a handle to a non-orientable surface is the same as adding two crosscaps).

Two invariants suffice to distinguish all compact surfaces:

• Orientability: whether all edge identifications preserve orientation.
• Euler characteristic: \(\chi = V - E + F\) for any polygonalization (invariant by Theorem 5.4). We have \(\chi((T^2)^{\#g}) = 2 - 2g\) and \(\chi((\mathbb{P}^2)^{\#h}) = 2 - h\).

Part III: The Fundamental Group

Chapter 6: Homotopy of Paths and the Fundamental Group

We now make the leap from point-set topology to algebraic topology: the programme of associating algebraic objects (groups, rings, modules) to topological spaces in a way that respects continuous maps. The most accessible such invariant is the fundamental group, which detects “holes” by studying loops.

Homotopy of paths

The key idea is that we should consider two paths “the same” if one can be continuously deformed into the other.

Think of \(s\) as a “time” parameter: at time \(s = 0\) we have the path \(\alpha\), and the family of paths \(f_s(t) = F(s, t)\) continuously deforms \(\alpha\) into \(\beta\) at time \(s = 1\), all while keeping the endpoints fixed.

The fundamental group

The verifications require explicit homotopies. For instance, \(\alpha \alpha^{-1} \simeq \kappa_a\): the homotopy goes “part way along \(\alpha\) and back,” going less and less far as the deformation parameter increases. Associativity \((\alpha\beta)\gamma \simeq \alpha(\beta\gamma)\) works by “reparametrizing” — the two paths traverse the same image, just at different speeds at different stages.

The fundamental group of a circle

The first non-trivial computation of a fundamental group is one of the most important results in all of topology.

The proof uses the technique of lifting paths to polar coordinates. Given a path \(\alpha: [0,1] \to \mathbb{C}^*\), one can uniquely lift it to a path \(\tilde{\alpha}(t) = (r(t), \theta(t))\) in \(\mathbb{R}^+ \times \mathbb{R}\) with \(\alpha(t) = r(t)e^{i\theta(t)}\), once an initial value \(\theta(0)\) is specified. The winding number \(\operatorname{wind}(\alpha, 0) = \frac{\theta(1) - \theta(0)}{2\pi}\) counts how many times \(\alpha\) wraps around the origin.

The key facts are:

1. Two loops at \(a\) in an annulus \(A = \{z \in \mathbb{C} \mid |z| \in I\}\) are homotopic if and only if they have the same winding number.
2. The loop \(\sigma^n(t) = ae^{2\pi i n t}\) has winding number \(n\).

The winding number therefore gives an isomorphism \(\pi_1(S^1, a) \to \mathbb{Z}\).

For those who know complex analysis, the winding number has an elegant integral formula: for a piecewise \(C^1\) loop \(\alpha\) in \(\mathbb{C} \setminus \{u\}\),

Functoriality

The fundamental group is not just an invariant of spaces — it is functorial, meaning it respects maps between spaces.

In the language of category theory: the fundamental group is a (covariant) functor from the category of based topological spaces to the category of groups.

Dependence on base point

If \(\gamma\) is a path from \(a\) to \(b\) in \(X\), the map \(\varphi_\gamma: \pi_1(X, a) \to \pi_1(X, b)\) given by \(\varphi_\gamma([\alpha]) = [\gamma^{-1} \alpha \gamma]\) is a group isomorphism. So in a path-connected space, the fundamental group is independent of the base point up to (non-canonical) isomorphism, and we may write simply \(\pi_1(X)\).

Chapter 7: Homotopy Invariance and Retracts

Homotopy equivalence of spaces

The fundamental group is invariant under an equivalence relation much weaker than homeomorphism.

Homotopy equivalence is dramatically coarser than homeomorphism. For instance, \(\mathbb{R}^n\) is homotopy equivalent to a point, and the Möbius band is homotopy equivalent to a circle. These spaces differ enormously in their metric and topological properties, but they share the same fundamental group.

Retracts and deformation retracts

If \(A\) is a deformation retract of \(X\), then \(A \simeq X\) and \(\pi_1(X, a) \cong \pi_1(A, a)\) for any \(a \in A\).

Applications of the fundamental group

The computation \(\pi_1(S^1) \cong \mathbb{Z}\) has remarkable consequences.

All of these classical results follow, directly or indirectly, from the non-triviality of \(\pi_1(S^1)\).

Part IV: Computing the Fundamental Group

Chapter 8: Free Groups and Free Products

To compute fundamental groups of complicated spaces, we need algebraic machinery: free groups and free products. These are the group-theoretic analogues of “putting things together without imposing relations.”

Direct products and sums

Recall that the direct product \(\prod_{k \in K} G_k\) consists of all functions \(a: K \to \bigcup G_k\) with \(a(k) \in G_k\), with componentwise multiplication. It satisfies a universal property for maps into it: a homomorphism to a product is determined by its components.

When the groups are abelian, the direct sum \(\bigoplus_{k \in K} G_k\) (the subgroup of elements with only finitely many non-identity components) satisfies the dual universal property for maps out of it.

Free products

For non-abelian groups, the correct “sum” construction is the free product, which plays the role of coproduct in the category of groups.

The free product satisfies the universal property: for any group \(H\) and homomorphisms \(f_k: G_k \to H\), there is a unique homomorphism \(f: \ast G_k \to H\) extending all the \(f_k\)’s, given by \(f(a_1 \cdots a_n) = f_{k_1}(a_1) \cdots f_{k_n}(a_n)\).

Free groups

Free groups are the “most general” groups generated by a set. Every group is a quotient of a free group: if \(A\) generates \(G\), then \(G \cong F(A)/N\) for some normal subgroup \(N\). This leads to presentations of groups.

The abelianization of a group \(G\) is \(\mathrm{Ab}(G) = G/[G, G]\), where \([G, G]\) is generated by commutators. The abelianization of a free group is a free abelian group: \(\mathrm{Ab}(F(A)) \cong \mathbb{Z}^A\).

Chapter 9: The Seifert-Van Kampen Theorem

The Seifert-Van Kampen theorem is the most powerful tool for computing fundamental groups. It computes \(\pi_1(X)\) from the fundamental groups of open subsets that cover \(X\).

In the common case of two open sets:

Applications

A deep consequence: every finitely presented group arises as the fundamental group of some topological space. Given \(G = \langle \alpha_1, \ldots, \alpha_n \mid w_1, \ldots, w_\ell \rangle\), form the wedge of \(n\) circles, then attach \(\ell\) discs along the words \(w_j\). The resulting CW complex has \(\pi_1 \cong G\).

Chapter 10: Covering Spaces

Covering spaces provide a geometric realization of the fundamental group. The theory reveals a beautiful dictionary between topology and group theory: subgroups of \(\pi_1(X)\) correspond to covering spaces of \(X\), just as subgroups of a group correspond to intermediate fields in Galois theory.

Covering spaces and covering maps

Path and homotopy lifting

The fundamental property of covering spaces is that paths and homotopies can be “lifted.”

Local path-connectedness

In a locally path-connected space, the connected components equal the path-components. Every manifold is locally path-connected (open balls in \(\mathbb{R}^n\) are convex, hence path-connected).

The classification of covering spaces

The proof constructs the universal cover explicitly: \(\tilde{X} = \{[\alpha] \mid \alpha \text{ is a path starting at } a\}\) with the topology generated by sets \(\tilde{U}_{[\alpha]} = \{[\alpha\lambda] \mid \lambda \text{ is a path in } U\}\) for elementary open sets \(U\). This is a beautifully concrete construction — points of the universal cover are homotopy classes of paths from the base point, and the covering map simply evaluates the endpoint.

For a general subgroup \(H\), one forms the quotient \(\tilde{X}_H = \tilde{X}/{\equiv}\) where \([\alpha] \equiv [\beta]\) if \(\alpha(1) = \beta(1)\) and \([\alpha\beta^{-1}] \in H\).

Part V: Beyond the Fundamental Group

Chapter 11: Algebraic Topology — Higher Invariants (PMATH 467)

The fundamental group \(\pi_1(X)\) is a remarkably effective invariant, but it has blind spots. It detects 1-dimensional holes — loops that cannot be contracted — but says nothing about higher-dimensional phenomena. The sphere \(S^2\) has \(\pi_1(S^2) = 0\), yet it is clearly not contractible (it encloses a 2-dimensional “hole”). To detect such features, we need more powerful machinery. PMATH 467: Algebraic Topology develops the full arsenal: higher homotopy groups, homology, cohomology, and the deep theorems they unlock.

Higher homotopy groups

The fundamental group studies maps from the circle \(S^1\) into \(X\). The natural generalization replaces the circle with higher-dimensional spheres.

For \(n = 1\), this recovers the fundamental group. For \(n \geq 2\), something remarkable happens: the group operation is abelian. Intuitively, this is because in two or more dimensions there is enough room to “slide” one map past another — a phenomenon that fails in one dimension, which is why \(\pi_1\) can be non-abelian.

The higher homotopy groups are functorial (a continuous map \(f: X \to Y\) induces homomorphisms \(f_*: \pi_n(X) \to \pi_n(Y)\)) and homotopy-invariant. A space is \(n\)-connected if \(\pi_k(X) = 0\) for all \(k \leq n\). Simply connected means 1-connected.

Computing \(\pi_n(S^k)\) for general \(n\) and \(k\) is one of the deepest unsolved problems in topology. The homotopy groups of spheres exhibit chaotic, seemingly unpredictable behaviour. Here is a small sample:

The pattern \(\pi_{n+k}(S^n)\) eventually stabilizes for large \(n\) (this is the Freudenthal suspension theorem); the resulting “stable” groups form the subject of stable homotopy theory, a central topic of modern algebraic topology.

Homology: counting holes algebraically

Homotopy groups are conceptually clean but fiendishly difficult to compute. Homology takes a different approach: instead of studying maps from spheres, it studies the space itself by decomposing it into simple pieces and tracking how those pieces fit together.

The key idea is the chain complex. One triangulates the space (or, more generally, decomposes it into cells) and forms free abelian groups \(C_n\) generated by the \(n\)-dimensional cells. The boundary operator \(\partial_n: C_n \to C_{n-1}\) records how each cell’s boundary is assembled from lower-dimensional cells. The crucial property is \(\partial_{n-1} \circ \partial_n = 0\) — “the boundary of a boundary is zero.”

Homology satisfies the Eilenberg-Steenrod axioms, which characterize it uniquely (up to a choice of coefficient group). Among the axioms, the most powerful is the Mayer-Vietoris sequence: if \(X = U \cup V\), there is a long exact sequence

This is the homological analogue of the Seifert-Van Kampen theorem, but it works in all dimensions and is often easier to apply.

The Euler characteristic from Chapter 5 appears here in its true generality:

For compact surfaces, this recovers the formula \(\chi = V - E + F\) from any triangulation.

The relationship between \(\pi_1\) and \(H_1\) is precise: \(H_1(X) \cong \pi_1(X)^{\text{ab}}\), the abelianization of the fundamental group. So homology forgets the non-abelian structure of \(\pi_1\) but gains computability and higher-dimensional reach.

Cohomology and the cup product

Where homology assigns groups, cohomology assigns dual groups — and gains a multiplicative structure.

The cup product is a strictly finer invariant than homology alone. A famous example: the spaces \(\mathbb{CP}^2\) and \(S^2 \vee S^4\) have identical homology groups, but their cohomology rings differ — \(H^*(\mathbb{CP}^2)\) has a non-trivial cup product (the square of the degree-2 generator is the degree-4 generator), while \(H^*(S^2 \vee S^4)\) has all cup products zero. The cup product detects how the “holes” in a space interact, not just how many there are.

The great theorems

The machinery of algebraic topology proves theorems that seem far removed from algebra.

The Brouwer Fixed Point Theorem (all dimensions). Every continuous map \(f: D^n \to D^n\) has a fixed point. The proof is a homological echo of our \(\pi_1\) argument: if \(f\) had no fixed point, we could construct a retraction \(r: D^n \to S^{n-1}\), which would induce a surjection \(H_{n-1}(D^n) \twoheadrightarrow H_{n-1}(S^{n-1})\). But \(H_{n-1}(D^n) = 0\) while \(H_{n-1}(S^{n-1}) \cong \mathbb{Z}\), a contradiction.

The Hairy Ball Theorem. There is no continuous non-vanishing tangent vector field on \(S^{2n}\) (even-dimensional spheres). You cannot comb a coconut without creating a cowlick. The proof uses the Euler class or an index-theoretic argument: a non-vanishing vector field would define a homotopy from the identity to the antipodal map, which has degree \((-1)^{2n+1} = -1 \neq 1\), contradicting the fact that homotopic maps have the same degree. On odd-dimensional spheres, non-vanishing vector fields do exist — for \(S^1\), the tangent vector field is simply rotation by 90 degrees.

The Borsuk-Ulam Theorem. For every continuous map \(f: S^n \to \mathbb{R}^n\), there exists a point \(x \in S^n\) with \(f(x) = f(-x)\). At every moment, some pair of antipodal points on Earth has the same temperature and the same barometric pressure. The proof for \(n = 2\) uses the fundamental group of \(\mathbb{RP}^2\); the general case uses \(\mathbb{Z}_2\)-cohomology. A delightful corollary:

The Ham Sandwich Theorem. Given \(n\) measurable sets in \(\mathbb{R}^n\), there exists a single hyperplane simultaneously bisecting all of them by volume. Two slices of bread and a slice of ham in \(\mathbb{R}^3\) can always be simultaneously halved by one cut — hence the name.

The Lefschetz Fixed Point Theorem. If \(f: X \to X\) is a continuous map of a compact triangulable space and the Lefschetz number \(\Lambda(f) = \sum_{n} (-1)^n \operatorname{tr}(f_*: H_n(X; \mathbb{Q}) \to H_n(X; \mathbb{Q}))\) is nonzero, then \(f\) has a fixed point. Brouwer’s theorem is the special case \(X = D^n\), where \(\Lambda(f) = 1\) for any \(f\). The Lefschetz theorem detects fixed points even when \(X\) has complicated topology.

Classification of Platonic Solids. There are exactly five convex regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The proof uses the Euler characteristic \(\chi(S^2) = V - E + F = 2\). If each face is a regular \(p\)-gon and \(q\) faces meet at each vertex, then double-counting gives \(qV = 2E = pF\), and substituting into \(\chi = 2\) yields \(\frac{1}{p} + \frac{1}{q} = \frac{1}{2} + \frac{1}{E}\). Since \(\frac{1}{p} + \frac{1}{q} > \frac{1}{2}\) and \(p, q \geq 3\), only five pairs \((p, q)\) work: \((3,3), (3,4), (4,3), (3,5), (5,3)\).

Deeper waters

Beyond these classical results, algebraic topology connects to many areas of modern mathematics:

• K-theory classifies vector bundles over spaces, linking topology to algebra and number theory. The Bott periodicity theorem says that the homotopy groups of the classifying spaces for vector bundles repeat with period 2 (complex) or 8 (real), a stunning rigidity.
• Characteristic classes (Stiefel-Whitney, Chern, Pontryagin) are cohomology classes associated to vector bundles that measure twisting. The non-orientability of the Möbius band, for instance, is detected by a non-trivial Stiefel-Whitney class.
• Spectral sequences are a bookkeeping device for computing homology from successive approximations. They are notorious for their complexity but indispensable for serious computations.
• The Poincare conjecture (proved by Perelman in 2003 using Ricci flow, not purely algebraic topology) states that every simply connected, closed 3-manifold is homeomorphic to \(S^3\). Higher-dimensional analogues were proved earlier by Smale (\(n \geq 5\)) and Freedman (\(n = 4\)) using surgery theory and cobordism — deep tools from algebraic topology.

Chapter 12: Topological Dynamics and Ergodic Theory (PMATH 457)

If algebraic topology studies the shape of spaces, topological dynamics studies what happens when you move around in them. The central question is: given a space \(X\) and a continuous self-map \(T: X \to X\), what can we say about the long-term behaviour of the orbits \(x, Tx, T^2x, T^3x, \ldots\)? Do they eventually cycle? Become dense? Distribute themselves evenly? PMATH 457 develops the theory in two intertwined halves: topological dynamics (using topology and combinatorics) and ergodic theory (adding measure theory).

Topological dynamical systems

The orbit of a point \(x\) is the set \(\mathcal{O}(x) = \{T^n x \mid n \geq 0\}\), and its closure \(\overline{\mathcal{O}(x)}\) is the orbit closure. The orbit closure is always a closed invariant set — a “sub-dynamical system” in its own right.

Minimality and recurrence

Minimal systems are the “irreducible” building blocks of topological dynamics. By Zorn’s Lemma (applied to the poset of non-empty closed invariant subsets), every compact dynamical system contains a minimal subsystem. The irrational rotation is the prototypical minimal system.

A point \(x\) is recurrent if it returns arbitrarily close to its starting position: for every neighbourhood \(U\) of \(x\), there exists \(n \geq 1\) with \(T^n x \in U\). In a minimal system, every point is recurrent. The Birkhoff recurrence theorem states that every compact dynamical system has at least one recurrent point — this is proved using Zorn’s Lemma and the intersection property of compact sets.

The Stone-Cech compactification and ultrafilters

One of the most profound ideas in topological dynamics is the use of ultrafilters and the Stone-Cech compactification \(\beta\mathbb{N}\) as a dynamical tool.

The set \(\beta\mathbb{N}\) of all ultrafilters on \(\mathbb{N}\), with the Stone topology, is a compact Hausdorff space in which \(\mathbb{N}\) embeds densely (via principal ultrafilters). The remarkable fact is that addition on \(\mathbb{N}\) extends to a semigroup operation on \(\beta\mathbb{N}\): define \(p + q = \{A \subseteq \mathbb{N} \mid \{n : A - n \in p\} \in q\}\). This operation is associative (but not commutative) and right-continuous. The algebraic structure of this semigroup — its idempotents (elements with \(p + p = p\)), minimal ideals, and smallest ideal — encodes deep combinatorial information about the integers.

From dynamics to combinatorics

The connection between topological dynamics and Ramsey theory is one of the jewels of modern combinatorics.

The original proof is purely combinatorial (and intricate). But there is a beautiful dynamical proof due to Furstenberg and Weiss. The idea is:

1. Encode the colouring as a point in a compact product space \(X = \{1, \ldots, r\}^{\mathbb{Z}}\) (the space of all bi-infinite sequences on \(r\) colours, with the product topology).
2. The shift map \(T: X \to X\) given by \((Tx)_n = x_{n+1}\) is a homeomorphism.
3. The colouring of \(\mathbb{Z}\) corresponds to a single point \(x_0 \in X\), and the orbit closure \(Y = \overline{\mathcal{O}(x_0)}\) is a compact invariant subsystem.
4. By the Birkhoff Multiple Recurrence Theorem (a non-trivial dynamical result), there is a point in \(Y\) that is “multiply recurrent,” which translates directly into a monochromatic arithmetic progression.

This approach generalizes vastly. The Hales-Jewett theorem — a much stronger result about “combinatorial lines” in the grid \(\{0, 1, \ldots, k-1\}^n\) for large \(n\) — also admits a proof via topological dynamics (though the standard proof is combinatorial). It implies Van der Waerden’s theorem as a special case.

Boundary actions and the greatest ambit

When a topological group \(G\) acts on a compact space \(X\), the dynamics encode information about the group itself. The greatest ambit of \(G\) is the “largest” compact \(G\)-flow with a distinguished point — it is the Stone-Cech compactification \(\beta G\) with the natural \(G\)-action, and every compact \(G\)-flow is a quotient of it.

Boundary actions arise when \(G\) acts on a compact space \(X\) and there is a \(G\)-fixed probability measure — the “boundary” is a minimal \(G\)-invariant subset that captures the asymptotic behaviour of orbits. For free groups and hyperbolic groups, the boundary is a Cantor set or a sphere, and its dynamics encodes the geometry of the group at infinity. This connects topological dynamics to geometric group theory.

Ergodic theory: adding measure

Topological dynamics tells us where orbits go, but not how often they visit different regions. For this, we need measures.

Ergodicity means the system cannot be decomposed into two independent subsystems of positive measure — it is “indecomposable” from the measure-theoretic viewpoint. The irrational rotation on \(S^1\) with Lebesgue measure is ergodic. The doubling map is ergodic. The identity map is not ergodic (every set is invariant).

The Krylov-Bogolyubov theorem guarantees that every continuous map on a compact metrizable space has at least one invariant probability measure — so the ergodic-theoretic viewpoint is always available.

The ergodic theorems

The mean ergodic theorem (von Neumann, 1932) and the pointwise ergodic theorem (Birkhoff, 1931) are the founding results of the subject. They say that time averages equal space averages.

In physical terms: if \(f\) measures some observable quantity (temperature, velocity, etc.) and \(T\) is the time evolution, then the long-run time average of \(f\) along almost every orbit equals the spatial average over the entire phase space. This is the mathematical justification for the ergodic hypothesis in statistical mechanics — the reason we can compute thermodynamic quantities by averaging over phase space instead of tracking individual particle trajectories.

Rokhlin’s Lemma: approximation by towers

The collection \(\{B, TB, \ldots, T^{n-1}B\}\) is called a Rokhlin tower of height \(n\). The lemma says that an aperiodic system can be approximated by a periodic one (the tower looks like a cyclic permutation) on a set of measure as close to 1 as desired. This is an indispensable tool for constructing examples and counterexamples in ergodic theory.

Entropy: measuring chaos

How “random” is a dynamical system? The concept of entropy, imported from information theory by Kolmogorov and Sinai, gives a precise numerical answer.

Entropy measures the rate at which the system generates information — how quickly you need new bits to describe the orbit of a typical point. A rotation (rational or irrational) has entropy 0: it is completely predictable. The doubling map \(x \mapsto 2x \mod 1\) has entropy \(\log 2\): each iteration effectively reveals one new binary digit.

Entropy is a conjugacy invariant: if two systems are measurably isomorphic, they have the same entropy. The converse fails in general, but for the most “random” systems it holds.

Ornstein’s theorem: the classification of Bernoulli shifts

The most spectacular application of entropy is the classification of Bernoulli shifts — the paradigmatic random systems.

This is a remarkable rigidity theorem. It says that for Bernoulli shifts, entropy is a complete invariant — it tells you everything. The “if” direction is the hard part: one must construct an explicit measure-preserving isomorphism between two product spaces with different alphabets but the same entropy. The proof introduces the concept of finitely determined processes and uses a deep approximation argument.

Ornstein’s theorem has a beautiful analogy with the classification of surfaces: just as compact surfaces are classified by two numbers (orientability and Euler characteristic), Bernoulli shifts are classified by a single number (entropy).

Orbit equivalence: dynamics beyond isomorphism

A further weakening of “same dynamics” asks only that orbits have the same structure, not the same map.

This stunning result says that from the point of view of orbit structure alone, all ergodic systems look the same. The entropy, which distinguishes Bernoulli shifts so precisely, is invisible to orbit equivalence. This shows that orbit equivalence is a much coarser relation than measurable isomorphism — it forgets the “speed” of the dynamics and retains only the “shape” of the orbits.

Dye’s theorem is the starting point for the deep connections between ergodic theory and the theory of von Neumann algebras in operator theory, where the orbit equivalence relation of a group action determines a factor of type \(\mathrm{II}_1\).

The web of connections

The interplay between the topological, combinatorial, and measure-theoretic viewpoints is the defining feature of PMATH 457. The subject draws on essentially everything developed in PMATH 367:

• Compact spaces are the stage for topological dynamics (Tychonoff’s theorem ensures that product spaces and Stone-Cech compactifications are available).
• Quotient spaces appear when forming orbit spaces and factor systems.
• Covering spaces relate to the dynamics of deck transformations.
• The fundamental group appears in the study of flows on surfaces and the dynamics of surface homeomorphisms.
• Product topologies underlie shift spaces and symbolic dynamics.

From this foundation, the subject radiates into number theory (equidistribution, continued fractions), combinatorics (Ramsey theory, additive combinatorics), probability (mixing, large deviations), and mathematical physics (statistical mechanics, quantum ergodicity).