Topology: Algebraic Topology

Algebraic topology studies shape using algebra. Instead of trying to classify spaces by drawing them or measuring them, it assigns algebraic objects such as groups, rings, and modules to topological spaces in a way that does not change under continuous deformation. These assignments produce topological invariants: properties that remain the same if you stretch, bend, or compress a space without tearing it or gluing new parts.

The payoff is practical as well as conceptual. Once a space is translated into algebra, many geometric questions become computable. You can prove that two spaces are not equivalent by showing their invariants differ, or confirm that certain maps must or must not exist because the algebra says so.

This article introduces four central pillars of the subject: the fundamental group, covering spaces, homology, and cohomology.

Why algebra is useful in topology

Topology focuses on properties preserved by continuous maps (homeomorphisms). A coffee mug and a doughnut are “the same” in this sense because each has one hole. But describing “holes” precisely is difficult if you rely only on pictures.

Algebraic topology formalizes holes at different dimensions:

• A loop around a missing point in the plane detects a 1-dimensional hole.
• A sphere surrounding a cavity detects a 2-dimensional hole.
• Higher-dimensional spaces can contain higher-dimensional analogues.

The main strategy is functorial: to a space $X$, assign an algebraic object $A(X)$, and to a continuous map $f:X\to Y$, assign a homomorphism $A(f):A(X)\to A(Y)$ that respects composition. This structure lets you compare spaces and maps systematically.

The fundamental group: loops as algebra

What it measures

The fundamental group ${\pi }_1(X,x_0)$ captures the ways you can loop around a space based at a point $x_0$. Consider all continuous loops $\gamma :[0,1]\to X$ with $\gamma (0)=\gamma (1)=x_0$. Two loops are considered equivalent if one can be continuously deformed into the other while keeping the basepoint fixed. This equivalence is called homotopy relative to endpoints.

Concatenating loops gives a group operation. The identity is the constant loop, and reversing a loop gives an inverse.

Examples that build intuition

• Simply connected spaces. If every loop can be shrunk to a point, then ${\pi }_1(X)$ is trivial. A sphere $S^2$ is simply connected.
• The circle. For $S^1$, loops can wind around any integer number of times. This yields

$${\pi }_1(S^1)\cong \mathbb{Z}.$$

• Punctured plane. ${\mathbb{R}}^2\setminus \{0\}$ has a hole, and its fundamental group is also $\mathbb{Z}$ because it deformation retracts onto a circle.

What it is good for

The fundamental group is often the first invariant used to distinguish spaces. If two spaces are homeomorphic, then their fundamental groups are isomorphic. In practice, showing ${\pi }_1(X)$ differs from ${\pi }_1(Y)$ is a clean way to prove $X$ and $Y$ are not topologically the same.

It also constrains maps. A continuous map $f:X\to Y$ induces a homomorphism $f_{\ast }:{\pi }_1(X)\to {\pi }_1(Y)$. If no homomorphism with required properties exists, then no such map exists either (at least not up to homotopy).

Covering spaces: local simplicity, global complexity

The basic idea

A covering space of $X$ is a space $\tilde{X}$ together with a continuous surjection $p:\tilde{X}\to X$ such that every point of $X$ has a neighborhood $U$ where $p^{-1}(U)$ is a disjoint union of open sets, each mapped homeomorphically onto $U$ by $p$. Locally, the cover looks like several identical “sheets” stacked over $U$.

Covering spaces are powerful because they turn global questions into local ones, where spaces are easier to handle.

Classic examples

• The real line covering the circle. The map $p:\mathbb{R}\to S^1$ given by $t\mapsto e^{2\pi it}$ wraps the line around the circle infinitely many times. Locally it is one-to-one, but globally it repeats.
• __MATH_INLINE_32__-fold covers of the circle. The map $z\mapsto z^n$ from $S^1$ to $S^1$ is an $n$-sheeted covering.
• Universal covers. For many “nice” spaces (path-connected, locally path-connected, semilocally simply connected), there exists a simply connected covering space called the universal cover.

The link to the fundamental group

Covering spaces and fundamental groups are tightly related. For a broad class of spaces, connected covering spaces correspond to subgroups of the fundamental group. Intuitively, ${\pi }_1(X)$ encodes how loops can lift to paths in $\tilde{X}$ and whether they close up.

This relationship makes coverings a computational tool: you can study ${\pi }_1(X)$ by analyzing its covers, and conversely classify covers by understanding subgroups of ${\pi }_1(X)$.

Homology: counting holes in all dimensions

Motivation and definition (conceptually)

While ${\pi }_1$ captures 1-dimensional looping behavior, it does not directly measure higher-dimensional holes and can be difficult to compute for complicated spaces. Homology provides a systematic and often computable family of invariants $H_n(X)$ for $n\ge 0$.

Very roughly, homology starts by approximating a space using building blocks (such as simplices in a triangulation). One forms formal combinations of $n$-dimensional pieces (called chains), identifies those that bound an $(n+1)$-dimensional piece (boundaries), and then defines $n$-dimensional homology as:

$$H_n(X)=\frac{\text{cycles in dimension }n}{\text{boundaries in dimension }n}.$$

The quotient captures cycles that do not bound, which correspond to $n$-dimensional holes.

Practical meaning through examples

• Connected components. $H_0(X)$ tracks connected components. If $X$ is connected, then $H_0(X)\cong \mathbb{Z}$.
• The circle. $H_1(S^1)\cong \mathbb{Z}$, reflecting the single 1-dimensional hole.
• The sphere. $H_2(S^2)\cong \mathbb{Z}$, capturing the enclosed 2-dimensional void; $H_1(S^2)=0$.
• The torus. A torus has independent 1-dimensional cycles running around it in two directions, which is reflected in $H_1$ having rank 2 (informally, two independent loop types). Its $H_2$ also detects the 2-dimensional “surface” class.

Even when you do not compute groups explicitly, homology gives invariants like Betti numbers, which are ranks of the free parts of $H_n(X)$ and serve as a robust measure of “how many $n$-dimensional holes” a space has.

Why homology is widely used

Homology is:

• Homotopy invariant: homotopy equivalent spaces have isomorphic homology groups.
• Computable: techniques like long exact sequences and Mayer-Vietoris (gluing spaces from pieces) make real calculations feasible.
• Stable: it behaves well under decomposition and construction.

Cohomology: duality and additional structure

From homology to cohomology

Cohomology assigns groups $H^n(X)$ to a space, often defined as algebraic duals of chain complexes used in homology. While homology is built from cycles and boundaries, cohomology is built from functions on chains (cochains) and their coboundaries.

Conceptually, if homology measures “holes,” cohomology measures “ways to detect holes.” A cohomology class can be evaluated on a homology class, producing an integer (or an element of another coefficient ring). This pairing connects geometry to computation.

The crucial extra feature: products

Cohomology is not just a list of groups. It typically carries a natural multiplication called the cup product, turning ${⨁}_n{H}^n(X)$ into a graded ring. This ring structure contains information that plain homology groups can miss.

For example, two spaces might have the same homology groups but different cohomology ring structures, allowing cohomology to distinguish them.

How it shows up in practice

Cohomology plays a central role in:

• Obstruction theory: deciding whether a map or a section of a bundle exists.
• Classification problems: distinguishing spaces beyond what homology can see.
• Geometry and analysis: cohomology classes can represent geometric data (such as differential forms in de Rham cohomology on smooth manifolds, where appropriate).

How these tools fit together

Algebraic topology is not a collection of isolated gadgets