Introduction to Algebraic Topology

Algebraic topology works by attaching an algebraic invariant to a space and then computing that invariant. The fundamental group measures one-dimensional holes. Homology measures holes of every dimension, and it does so in a form where the two halves of the story — “chains” and “cycles” — are actual algebraic objects you can manipulate. The entire subject is the observation that the pieces fit: the fundamental group is $H_1$ with extra structure, and the long exact sequence of a pair is what makes the two fit together.

The progression below is deliberate. Sections 9.1 to 9.5 build the fundamental group from scratch and give the one theorem (van Kampen) that lets you compute it for a space assembled from pieces. Sections 9.6 to 9.9 introduce chains, cycles and boundaries, then homology, then the two tools that make homology usable — the long exact sequence and Mayer-Vietoris — and then the duality and cohomology that close the subject. Every computation is worked in full.

Picture a doughnut. Walk a loop around the hole: it does not contract, and that failure is recorded by an integer — how many times you went round. Now put a second handle on the doughnut and ask the same question: you need two integers. Now ask about a void rather than a handle: a loop tells you nothing, because a void is not something a loop can detect. That is the limitation that motivates homology: the fundamental group sees tunnels but is blind to cavities, and it does so because a loop is a one-dimensional object and a cavity is three-dimensional.

Homology fixes this by letting the investigator choose the dimension. $H_k$ is built from $k$-dimensional chains, so $H_1$ sees tunnels, $H_2$ sees the middle sphere of a double doughnut, and $H_3$ sees cavities. The price is that the fundamental group’s non-abelian information is lost — and it is genuinely lost, not hidden, which is why both theories exist rather than one replacing the other. The relationship is exact: the abelianisation of $\pi_1(X)$ is $H_1(X)$.

The second intuition is that a boundary is a degeneracy. A triangle bounds a filled-in region; a loop is a triangle whose filling is missing. Homology is built by declaring “everything that bounds something is zero” — the quotient of chains by boundaries. What survives the quotient is the thing with no filling, which is precisely the hole.

Definition. Two continuous maps $f, g : X \to Y$ are homotopic (written $f \simeq g$) if there is a continuous $H : X \times [0,1] \to Y$ with $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x$.

Definition. Spaces $X$ and $Y$ are homotopy equivalent ($X \simeq Y$) if $f : X \to Y$ and $g : Y \to X$ satisfy $g \circ f \simeq \operatorname{id}_X$ and $f \circ g \simeq \operatorname{id}_Y$.

Example. $D^2 = \{(x,y) : x^2 + y^2 \leq 1\}$ is homotopy equivalent to a point — it is contractible, via $H(x, t) = (1-t)x$.

Homotopy equivalence is much weaker than homeomorphism: $\mathbb{R}^n \simeq \{*\}$ but the two are not homeomorphic. It is still an equivalence relation, and concordance classes of maps between two fixed spaces are exactly the homotopy classes of maps — which is what makes $\pi_1$ well defined.

Definition. A loop in $X$ based at $x_0$ is a continuous $\gamma : [0,1] \to X$ with $\gamma(0) = \gamma(1) = x_0$.

Definition. $\pi_1(X, x_0)$ is the set of homotopy classes of loops based at $x_0$, under concatenation. Reversal of orientation is inversion, and the identity is the class of the constant loop. For path-connected $X$ the group is independent of $x_0$ up to isomorphism.

The group law is concatenation of representatives, and this is the step where students routinely get it wrong: the product $[\gamma][\delta]$ is represented by the loop that traces $\gamma$ and then $\delta$. It is well defined on classes precisely because concatenation respects homotopy.

Example. $\pi_1(S^1) \cong \mathbb{Z}$, and the isomorphism is the winding number. Given a loop $\gamma$ in $S^1$, lift it to $\mathbb{R}$ via $p(t) = e^{2\pi i t}$; because $\gamma$ closes up, the lift satisfies $\tilde{\gamma}(1) = \tilde{\gamma}(0) + n$ for a unique $n \in \mathbb{Z}$, and $n$ is the winding number. Homotopic loops have the same $n$ because a homotopy lifts to a homotopy in $\mathbb{R}$, which keeps the endpoint difference fixed.

Example. $\pi_1(\mathbb{R}^n) = \{e\}$ for all $n \geq 1$, since $\mathbb{R}^n$ is convex: for a loop $\gamma$ use $H(x, t) = \gamma(tx)$, which contracts any path to its basepoint.

Example. $\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$. Loops are classified by a pair of winding numbers, one for each generator of the torus’s fundamental group.

Example. $\pi_1(S^1 \vee S^1)$ is the free group $F_2$ on two generators, and is not abelian. This is the standard warning: $\pi_1$ is a genuine non-abelian invariant, unlike $H_1$ below.

Definition. A path-connected space $X$ is simply connected if $\pi_1(X) \cong \{e\}$, that is, if every loop contracts to a point.

Proposition. $\mathbb{R}^n$, $S^n$ for $n \geq 2$, and every convex subset of $\mathbb{R}^n$ are simply connected.

The assertion about $S^n$ for $n \geq 2$ is a theorem (the sphere is $n$-connected for $n \geq 2$) and not a consequence of the convexity argument, which is worth noticing: $S^n$ is not convex, and the failure of the naive argument is what makes the result interesting.

Definition. A continuous surjection $p : \tilde{X} \to X$ is a covering map if every $x \in X$ has a neighbourhood $U$ with $p^{-1}(U) = \bigsqcup_i \tilde{U}_i$, each $\tilde{U}_i \mapsto U$ a homeomorphism.

Example. $\mathbb{R} \to S^1$, $t \mapsto e^{2\pi i t}$, is a universal cover: the preimage of any small arc is a disjoint union of open intervals.

Theorem (Lifting). If $p : \tilde{X} \to X$ is a covering, $\tilde{X}$ is path-connected and locally path-connected, $Y$ is locally path-connected, $\tilde{y}_0 \in p^{-1}(y_0)$ and $f : Y \to X$ a map with $f(y_0) = p(\tilde{y}_0)$, then $f$ lifts uniquely to $\tilde{f} : Y \to \tilde{X}$.

This theorem is what makes coverings computable. Take a loop $\gamma$ at $y_0$ and lift it starting from $\tilde{y}_0$; where the lift ends records the element of $\pi_1$. For $\mathbb{R} \to S^1$ the endpoint is $\tilde{y}_0 + n$, and $n$ is the winding number — which is the proof of $\pi_1(S^1) \cong \mathbb{Z}$ promised in 9.2, and the reason coverings and the fundamental group are the same subject viewed twice.

You cannot compute $\pi_1$ of an arbitrary space directly. van Kampen is what makes the fundamental group practical: it computes it from a decomposition of the space into two open sets you already understand.

Theorem (Seifert—van Kampen). Let $X = U \cup V$ with $U, V$ open and path-connected, $U \cap V$ path-connected, and let $x_0 \in U \cap V$. Then $\pi_1(X, x_0)$ is the pushout

that is, the quotient of the free product $\pi_1(U) * \pi_1(V)$ by the normal closure of $\{(i_U(\gamma), i_V(\gamma)^{-1}) : \gamma \in \pi_1(U \cap V)\}$, where the inclusions into $U$ and $V$ are induced by a path from $x_0$ into each set. If $U \cap V$ has two or more components, add generators for those components and no relations.

Worked Example. Let $X = S^1 \vee S^1$ (the figure eight). Take $U = S^1 \vee \{*\}$ and $V = \{*\} \cup S^1$, each a full copy of $S^1$, and $U \cap V$ contractible. Then

with no relations to impose, because $\pi_1(U \cap V)$ is trivial. This is the mechanism behind every familiar computation: gluing two spaces along something simply connected produces a free product.

Worked Example. A genus-$g$ orientable surface $\Sigma_g$ is a sphere with $g$ handles attached. Remove a small disc from each of the $g$ handles and let $U$ be the sphere with $g$ holes, $V$ the union of the $g$ discs. Each boundary circle is a generator of $\pi_1(U)$, and $U$ deformation-retracts onto a wedge of $g$ circles, so $\pi_1(U) = F_g$; $V$ is a disjoint union of contractible discs; $U \cap V$ is $g$ disjoint circles, so $\pi_1(U \cap V) = \mathbb{Z}^g$. Van Kampen with several components imposes no relations, hence

the torus, recovering 9.2. Notice that $\Sigma_2$ has $\pi_1 = F_2$, the same as the figure eight, but $\Sigma_2$ is not homeomorphic to it: $\Sigma_2$ has a 2-dimensional hole and the figure eight does not. $\pi_1$ cannot see that difference. That is the precise limitation homology was invented to remove.

Everything up to here has measured one-dimensional holes. To measure holes of other dimensions, the algebra has to be built from scratch, and it is built by mirroring the definition of singular $k$-simplices on the geometric picture.

Definition. A singular $k$-simplex is a continuous map $\sigma : \Delta^k \to X$, where $\Delta^k \subset \mathbb{R}^{k+1}$ is the standard $k$-simplex. Write $C_k(X)$ for the free abelian group on all singular $k$-simplices; elements are finite formal sums $\sum_i n_i \sigma_i$, called $k$-chains.

Definition. The boundary of a simplex $\sigma$ is $\partial \sigma = \sum_{i=0}^{k} (-1)^i \left( \sigma \big|_{F_i} \right)$, where $F_i$ is the $(k-1)$-face of $\Delta^k$ obtained by deleting vertex $i$ and $\sigma|_{F_i}$ is its restriction, itself a singular $(k-1)$-simplex. This extends to a homomorphism $\partial_k : C_k(X) \to C_{k-1}(X)$, and $C_{-1}(X) = 0$.

Lemma. $\partial^2 = 0$.

Proof. Each $(k-1)$-face of $\sigma$ is counted twice in $\partial(\partial\sigma)$, with opposite signs, by the parity rule. Explicitly, a $(k-2)$-face $F$ of $\Delta^k$ is contained in exactly two $(k-1)$-faces, namely those deleting $i$ and $j$ for the two vertices not in $F$; its coefficients are $(-1)^i$ and $(-1)^j$ with opposite parity. The terms cancel in pairs. $\blacksquare$

Definition. A $k$-chain $c$ is a cycle if $\partial c = 0$. The subgroup of cycles is $Z_k(X)$. A $k$-chain $c$ is a boundary if $c = \partial d$ for some $(k+1)$-chain $d$; these form $B_k(X)$.

Definition. The $k$-th singular homology group is

Two cycles that differ by a boundary represent the same class. The geometric reading is the whole point: a cycle is a closed surface made of simplices, and it is trivial in homology exactly when it bounds a solid region. A hole is a closed thing with no filling.

Example. $H_0(X;\mathbb{Z}) \cong \mathbb{Z}^{[X^0]}$, a free abelian group on the path components, where $X^0$ is the set of components. Each point is its own 0-simplex, and it is a cycle vacuously, and it is a boundary only if it is the boundary of a path to another point. So $H_0$ counts components — the zeroth homology group is the first thing the construction gets right.

Example. $H_1(S^1) \cong \mathbb{Z}$, $H_1(T^2) \cong \mathbb{Z}^2$, $H_1(\Sigma_g) \cong \mathbb{Z}^{2g}$, and $H_k(S^n) = 0$ for $k \neq n$ with $H_n(S^n) = \mathbb{Z}$. The dimension-$n$ generator is the sphere itself.

Example. $H_2(\Sigma_g) = 0$ for every $g$, including the torus. This is the statement that a torus has no two-dimensional hole: cutting it open at one point leaves a disc, which is contractible. Contrast $H_2(S^2) = \mathbb{Z}$ and $H_2(\Sigma_2) = \mathbb{Z}$, where the two handles together bound a middle surface.

Theorem. For any path-connected space $X$, $H_1(X;\mathbb{Z})$ is the abelianisation of $\pi_1(X)$:

where $[\pi_1, \pi_1]$ is the commutator subgroup.

Sketch. Homotopy classes of loops map to $1$-cycles by $\gamma \mapsto \gamma$, a 1-simplex traversed in order. Concatenation of loops becomes addition of chains. This map is surjective onto $H_1$ and its kernel is exactly the commutator subgroup. $\blacksquare$

So the two theories agree on abelian information and $H_1$ throws away exactly the non-abelian part. For $S^1 \vee S^1$ this gives $H_1 = \mathbb{Z}^2$ from the free group $F_2$ — the same group, but now it is also the homology of $T^2$, and homology has a computation machinery (Mayer- Vietoris, excision) that the fundamental group does not. That is the trade.

The long exact sequence is what makes homology usable rather than merely defined. It lets you compute an unknown group from ones you can compute.

Theorem (Long exact sequence of a pair). If $A \subseteq X$ and $H_*(\partial A) \to H_*(A) \to H_*(X)$ then

is exact, where $H_k(X, A) = H_k(X, A)/H_k(A)$ are the relative groups. Exactness at each term means the image of each map equals the kernel of the next.

Theorem (Mayer-Vietoris). If $X = A \cup B$ with $A$ and $B$ having intersection of the homotopy type of a finite disjoint union of circles, then $\cdots \to H_n(A \cap B) \xrightarrow{\phi} H_n(A) \oplus H_n(B) \xrightarrow{\psi} H_n(X) \to H_{n-1}(A \cap B) \to \cdots$

is exact, where $\phi$ is $(i_{A*}, -i_{B*})$ and $\psi(a, b) = i_{A*}a + i_{B*}b$.

Worked Example. Compute $H_*(S^2)$. Write $S^2 = D_+ \cup D_-$ as two closed hemispheres. Each hemisphere is contractible, so $H_0(D_\pm) = \mathbb{Z}$ and $H_n(D_\pm) = 0$ for $n \geq 1$. The intersection is the equator $S^1$, with $H_0 = \mathbb{Z}$ and $H_1 = \mathbb{Z}$.

For $n \geq 2$ the sequence reads $0 \to 0 \to H_n(S^2) \to 0 \to 0$, so $H_n(S^2) = 0$.

For $n = 1$:

which is

The map $(1,-1)$ is injective, so exactness forces $H_1(S^2) = 0$. For $n = 0$ the same sequence gives

and $\operatorname{coker}(1,1) = \mathbb{Z}$, since $(1,1)$ is injective and its cokernel is $\mathbb{Z}^2/\langle(1,1)\rangle \cong \mathbb{Z}$. Hence $H_0(S^2) = \mathbb{Z}$: one component, as it must be.

Collecting: $H_k(S^2) = \mathbb{Z}$ for $k = 0, 2$ and $0$ otherwise. $\blacksquare$

Worked Example. Compute $H_*(\Sigma_2)$, the double torus, which has no closed form and no easy deformation retract. Let $A$ be the sphere, $B$ two discs, so $\Sigma_2 = A \cup B$. Then $\pi_1$-style, $A$ retracts onto a wedge of two circles $F_2$, so $H_1(A) = \mathbb{Z}^2$ and $H_0(A) = \mathbb{Z}$; $B$ is a disjoint union of two discs, so $H_0(B) = \mathbb{Z}^2$, $H_n(B) = 0$ for $n\geq1$; and $A \cap B$ is two circles, so $H_1 = \mathbb{Z}^2$, $H_0 = \mathbb{Z}^2$.

At $n = 2$:

and $\phi : \mathbb{Z}^2 \to \mathbb{Z}^2 \oplus 0$ sends the two boundary circles to the two generators of $F_2$, so it is injective. Exactness gives $H_2(\Sigma_2) = 0$.

At $n = 1$:

is

The first map is injective, so $H_1(\Sigma_2) = \operatorname{coker}\,(1,-1) = \mathbb{Z}^4 / \langle (1,-1) \rangle \cong \mathbb{Z}^3$, since $(1,-1)$ is primitive and the quotient of $\mathbb{Z}^2$ by a primitive rank-one submodule is $\mathbb{Z}$. And the last map is injective, so nothing further is contributed. Hence

matching $\chi = 2 - 2g = -2 = 1 - 3$. $\blacksquare$

Definition. The $k$-th singular cohomology group is the dual of homology:

for an abelian group $G$. A class $\varphi \in H^k(X; G)$ is a homomorphism $\varphi : H_k(X) \to G$, and evaluating it on a cycle is a measurement of that cycle. So cohomology measures holes while homology counts them, and for finite complexes the two are related by a universal coefficient theorem:

which, for $G = \mathbb{Z}$ and $H_{k-1}$ free, gives $H^k(X;\mathbb{Z}) \cong \operatorname{Hom}(H_k(X), \mathbb{Z})$.

The reason to use cohomology anyway is that it carries a product: there are maps $\smile : H^p \times H^q \to H^{p+q}$, the cup product, and the theorem that made cohomology indispensable is that they detect structure homology cannot.

Theorem. $S^2 \not\simeq \mathbb{R}P^2$.

Proof. $\mathbb{R}P^2$ has $H_1(\mathbb{R}P^2) = \mathbb{Z}/2$ — its fundamental group is $\mathbb{Z}/2$ and by 9.7 so is its $H_1$ — whereas $H_1(S^2) = 0$ by the computation in 9.8. Since homology is a homotopy invariant, the two spaces are not homotopy equivalent. $\blacksquare$

The stronger statement is that the cohomology ring separates spaces homology cannot. Take $\mathbb{R}P^2$ and $S^2 \vee S^2$: both have the same homology groups, both have $H^1 \cong 0$, but the cup square of the generator of $H^1(\mathbb{R}P^2; \mathbb{Z}/2)$ is the generator of $H^2(\mathbb{R}P^2; \mathbb{Z}/2)$, whereas all products of degree-one classes on $S^2 \vee S^2$ vanish. The ring remembers what the groups forget. $\blacksquare$

Definition. For a finite CW-complex, the Euler characteristic is $\chi(X) = \sum_k (-1)^k c_k$, where $c_k$ is the number of $k$-cells. For a triangulated surface this is $V - E + F$.

Theorem. $\chi$ is a homotopy invariant, and for a finite CW-complex $\chi(X) = \sum_k (-1)^k \operatorname{rank} H_k(X)$.

For a closed orientable surface of genus $g$, $\chi = 2 - 2g$; for a non-orientable one, $\chi = 2 - g$. The odd/even signature of $g$ is exactly what separates the two families, which is why the classification of surfaces reduces to one integer once orientability is known.

Theorem (Classification of compact surfaces). Every compact connected surface is homeomorphic to exactly one of: a sphere with $g$ handles (orientable, genus $g$), or a sphere with $g$ cross-caps (non-orientable, genus $g$).

• Torus $T^2$ — a square with opposite edges identified in the same direction; $\pi_1 = \mathbb{Z}^2$.
• Projective plane $\mathbb{R}P^2$ — $S^2$ with antipodal points identified; non-orientable, $\pi_1 = \mathbb{Z}/2$.
• Klein bottle — a square with one pair of edges reversed; non-orientable, and cannot be embedded in $\mathbb{R}^3$.

The classification is the payoff of the whole apparatus: an infinite-looking family of surfaces is indexed by a single integer.

• Homotopy equivalence is weaker than homeomorphism. $\mathbb{R}^n \simeq \{*\}$, yet the two are not homeomorphic.
• $\pi_1$ detects one-dimensional holes only. $\Sigma_2$ and $S^1 \vee S^1$ have the same fundamental group $F_2$ but are not homeomorphic, because $\Sigma_2$ has a 2-dimensional hole and $H_2(\Sigma_2) = 0$ records its absence.
• $H_1$ is the abelianisation of $\pi_1$. Homology discards exactly the non-abelian information, and gains a computation machinery in exchange.
• Euler characteristic is a homotopy invariant and equals the alternating sum of homology ranks, so it can be computed either way.
• Coverings relate local and global topology. The universal cover is simply connected, and $\pi_1$ acts on it by deck transformations, so $\pi_1(X)$ is literally the symmetry group of the cover.
• Cohomology with products separates spaces homology cannot. The ring structure on $H^*(\mathbb{R}P^2; \mathbb{Z}/2)$ distinguishes it from $S^2 \vee S^2$, which have identical homology groups.
• Van Kampen computes $\pi_1$ the way Mayer-Vietoris computes homology. Both take a decomposition and combine known pieces, and both are exact sequences.

• Confusing homotopy with homeomorphism. A solid disc and a point are homotopy equivalent but not homeomorphic. Homotopy equivalence is a much coarser relation.
• Assuming $\pi_1(X)$ is always abelian. $\pi_1(S^1 \vee S^1) = F_2$ is not. This is precisely the information that $H_1$ discards.
• Forgetting that $\pi_1$ depends on the basepoint. For a space that is not path-connected, different components can have different fundamental groups.
• Conflating “cycle” with “class”. A cycle is a chain with $\partial c = 0$; a homology class is a cycle modulo the boundaries. Every nontrivial-looking cycle may still be trivial in homology if it bounds.
• Expecting $\partial^2 = 0$ to be automatic. It is a theorem, and it is what makes $H_k$ well defined. The sign parity in the boundary formula is doing real work: without it, every face would be counted twice with the same sign and the identity would fail.
• Applying Mayer-Vietoris without checking exactness. At each step you must take the kernel of one map and the cokernel of the previous; skipping a term gives the wrong rank.
• Using $H_k$ to detect cavities you cannot triangulate. $H_3$ detects cavities in dimension 3, but the cavity must actually bound nothing in the space you are computing.
• Claiming $\chi$ distinguishes all surfaces. It does not: the torus and the Klein bottle both have $\chi = 0$ and are not homeomorphic. Orientability is needed as well.

• Robotics and motion planning. The fundamental group of a configuration space determines whether two configurations can be continuously deformed into each other; homotopy classes of paths are the configuration space’s components.
• Topological data analysis. Persistent homology computes the homology of a filtration of a data set, producing a barcode whose bars are the connected components, loops and voids that persist across scales.
• Condensed matter physics. Homotopy groups classify topological defects: vortices in superfluids and dislocations in crystals are $\pi_1$ defects, and the order parameter’s winding is measured by $\pi_1$ of the vacuum manifold.
• Knot theory. The knot group $\pi_1(S^3 \setminus K)$ is a powerful knot invariant, distinguishing knots that share crossing number.
• Algebraic geometry. Poincare duality relates the cohomology of a smooth projective variety to its homology, and the Lefschetz hyperplane theorem is a deep restriction of that relationship.

• Metric Spaces — the metric structure underpins the continuity and convergence used throughout.
• Closed Sets, Closure, Interior, and Boundary — open and closed sets are what make covering spaces and CW-complexes well defined.
• Compactness — every finite CW-complex is compact, which is what makes the Euler characteristic a homotopy invariant.
• Separation Axioms — the hypotheses of the lifting theorem and van Kampen.
• Sequences and Limits — convergence is the analytic shadow of continuity.
• Homomorphisms and Isomorphism Theorems — the quotient by a normal subgroup is the same construction that makes $H_k$ and $\pi_1/[\pi_1,\pi_1]$ well defined.
• Quantum Mechanics — symmetry groups, which is what the fundamental group of a configuration space is made of.
• Classical Mechanics

• Closed Sets, Closure, Interior, and Boundary
• Common Pitfalls
• Compactness
• Connectedness