Algebraic Topology

1. Topological Spaces and Homeomorphisms

Algebraic topology begins with the category of topological spaces and continuous maps. A topological space is a set X equipped with a topology: a collection of subsets called open sets that is closed under arbitrary unions and finite intersections and that contains both the empty set and X itself. Every metric space is a topological space, but topologies can exist without any metric.

Homeomorphisms

Two topological spaces are homeomorphic when there exists a bijective continuous map between them whose inverse is also continuous. Homeomorphic spaces are topologically identical: they share every topological property. A coffee mug is homeomorphic to a donut because both have exactly one hole, even though their geometric shapes differ.

Homotopy Equivalence

Homotopy equivalence is coarser than homeomorphism. Two spaces X and Y are homotopy equivalent when there are continuous maps f from X to Y and g from Y to X such that the composition g after f is homotopic to the identity on X and f after g is homotopic to the identity on Y. A homotopy between two maps is a continuous one-parameter family of maps interpolating between them.

The key insight is that algebraic invariants such as homotopy groups and homology groups are homotopy invariants: they depend only on the homotopy type, not the precise homeomorphism class. This makes homotopy equivalence the natural equivalence relation for most of algebraic topology.

Deformation Retracts

A subspace A of X is a deformation retract of X when there is a homotopy from the identity map on X to a map that sends all of X into A while fixing every point of A throughout the homotopy. Deformation retracts give the most common way to exhibit homotopy equivalences in practice. For example, the unit circle is a deformation retract of the punctured plane: each point other than the origin is retracted radially toward the circle while the circle itself stays fixed.

2. The Fundamental Group

The fundamental group is the first and most important algebraic invariant in algebraic topology. It encodes the one-dimensional connectivity of a space by recording how loops can be continuously deformed into one another.

Definition

Fix a basepoint x in a topological space X. A loop based at x is a continuous map from the unit interval into X that sends both endpoints to x. Two loops are homotopic relative to the basepoint when one can be continuously deformed into the other while keeping the basepoint fixed throughout. The fundamental group, written pi sub 1 of X at x, is the set of homotopy classes of loops, with the group operation given by path concatenation: traverse the first loop, then traverse the second.

Simply Connected Spaces

A path-connected space is simply connected when its fundamental group is trivial, meaning every loop is homotopic to the constant loop. Simply connected spaces have no one-dimensional holes. Euclidean space, the two-sphere, and any convex subset of Euclidean space are simply connected.

Fundamental Group of the Circle

The fundamental group of the circle is isomorphic to the integers. The generator is the loop that winds once counterclockwise around the circle. Winding n times counterclockwise represents the integer n, and winding n times clockwise represents negative n. The group operation of concatenation corresponds to addition of integers: winding twice then winding three times is the same as winding five times. The integer associated to a loop is its winding number.

Fundamental Group of the Torus

The torus is the product of two circles. Its fundamental group is the direct product of the fundamental groups of its factors, which is the group of pairs of integers under componentwise addition. Geometrically, there are two independent families of loops: those that go around the longitude of the torus and those that go around the meridian. Because the torus is a product, these loops commute with each other, giving an abelian fundamental group.

Fundamental Group of the Two-Sphere

The two-sphere is simply connected: its fundamental group is trivial. Every loop on the sphere can be contracted to a point because the sphere has no one-dimensional holes — you can always slide a loop off to the side and shrink it down. This contrasts sharply with the circle, where the winding number prevents nontrivial loops from contracting.

Fundamental Group of Real Projective Space

The real projective plane is obtained from the two-sphere by identifying each pair of antipodal points. Its fundamental group is the group of two elements, written Z mod 2. The nontrivial loop travels from a point to its antipode, which has been identified with the starting point, and this loop cannot be contracted. Traversing it twice gives a loop that can be contracted, so the element has order 2.

Induced Homomorphisms

A continuous map f from X to Y sends loops in X to loops in Y and sends homotopies to homotopies, so it induces a group homomorphism from the fundamental group of X to the fundamental group of Y. This makes the fundamental group a functor from the category of pointed topological spaces to the category of groups. Homeomorphic spaces have isomorphic fundamental groups, and homotopy equivalent spaces also have isomorphic fundamental groups.

3. Van Kampen's Theorem and Free Products

Van Kampen's theorem is the main computational tool for fundamental groups. It expresses the fundamental group of a space assembled from simpler pieces in terms of the fundamental groups of the pieces and their overlap.

Free Products of Groups

The free product of two groups G and H is the group whose elements are finite alternating strings of elements from G and H, with the group operation given by concatenation and simplification. Unlike the direct product, elements from G and H do not commute in the free product unless forced to. The free product is the coproduct in the category of groups.

The Theorem Statement

Suppose X is the union of two path-connected open subsets A and B whose intersection is also path-connected, with basepoint x in the intersection. Then the fundamental group of X is the free product of the fundamental group of A and the fundamental group of B, amalgamated over the fundamental group of A intersect B, where the amalgamation is performed via the homomorphisms induced by the inclusions of A intersect B into A and into B.

Applications

Abelianization and the First Homology

The first homology group of a path-connected space is the abelianization of its fundamental group: take the fundamental group and impose the additional relations that all elements commute with each other. For the wedge of two circles, the fundamental group is the free group on two generators (nonabelian), and its abelianization is the free abelian group on two generators, which is a product of two copies of the integers.

4. Covering Spaces and Lifting Properties

Covering space theory translates topological questions about spaces into algebraic questions about groups and their subgroups, giving one of the deepest connections between topology and algebra.

Definition of a Covering Space

A covering space of a space X is a space E together with a continuous surjective map p from E to X such that for every point in X there is an open neighborhood U for which the preimage of U under p is a disjoint union of open sets in E, each mapped homeomorphically onto U by p. The map p is called the covering map and the preimage of each point is called the fiber.

Path Lifting and Homotopy Lifting

Covering maps satisfy two key lifting properties. First, any path in X starting at a point x can be lifted uniquely to a path in E starting at any chosen point in the fiber above x. Second, any homotopy of paths in X lifts to a homotopy of paths in E with matching initial conditions. These lifting properties give precise control over how loops in X behave when pulled back to E.

The Classification Theorem

For a path-connected, locally path-connected, and semilocally simply connected space X with basepoint x, there is a one-to-one correspondence between connected covering spaces of X (up to isomorphism over X) and subgroups of the fundamental group of X at x. The universal cover corresponds to the trivial subgroup and has trivial fundamental group. Normal subgroups correspond to regular covering spaces whose deck transformations act transitively on each fiber.

Deck Transformations

A deck transformation of a covering space E over X is a homeomorphism of E to itself that commutes with the covering map p. The deck transformations form a group. For the universal cover, the deck transformation group is isomorphic to the fundamental group of X: the topology of the base space is completely encoded in the symmetries of its universal cover. For the real line covering the circle, the deck transformations are the integer translations, and this group of translations is isomorphic to the integers, which is the fundamental group of the circle.

5. Homology: Simplicial and Singular

Homology theory assigns a sequence of abelian groups to a topological space, measuring the presence of holes in each dimension. Unlike the fundamental group, homology groups are always abelian, making them easier to compute and classify.

Simplices and Simplicial Complexes

An n-simplex is the convex hull of n plus 1 points in general position in Euclidean space. A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. A simplicial complex is a finite collection of simplices glued together along their faces, forming a piecewise linear topological space.

Chain Complexes and Boundary Maps

Given a simplicial complex, the n-th chain group is the free abelian group generated by the oriented n-simplices. The boundary map sends each n-simplex to the alternating sum of its faces, with signs determined by orientation. The fundamental identity of homology theory is that the boundary of a boundary is zero: any face of a boundary has been counted twice with opposite signs and cancels out.

The chain complex is the sequence of chain groups connected by boundary maps. An n-cycle is a chain with zero boundary; it represents a closed n-dimensional loop or surface. An n-boundary is a chain that is the boundary of an (n plus 1)-chain; it represents something that fills in. The n-th simplicial homology group is the quotient of the group of n-cycles by the subgroup of n-boundaries.

Singular Homology

Singular homology replaces the rigid combinatorial simplices with flexible continuous maps. A singular n-simplex in a space X is any continuous map from the standard n-simplex into X. The singular chain group in degree n is the free abelian group on all singular n-simplices. The boundary map is defined by the alternating sum of face maps, and the singular homology groups are computed as cycles modulo boundaries just as in the simplicial case.

Singular homology is defined for any topological space, not just those with a simplicial structure. For spaces that admit a triangulation, singular and simplicial homology agree. Singular homology is functorial: every continuous map induces homomorphisms on singular homology groups, and homotopic maps induce the same homomorphisms.

Homology of Standard Spaces

Reduced Homology and Long Exact Sequences

Reduced homology modifies the zeroth homology group to be zero for a nonempty connected space, making many formulas cleaner. The long exact sequence of a pair (X, A), where A is a subspace of X, connects the homology groups of A, X, and the relative homology groups of the pair in a long exact sequence, giving a systematic way to compute homology groups inductively.

6. The Mayer-Vietoris Sequence

The Mayer-Vietoris sequence is the homological analogue of van Kampen's theorem. It expresses the homology of a space X written as the union of two open sets A and B in terms of the homology of A, B, and their intersection.

The Sequence

The Mayer-Vietoris sequence is a long exact sequence of homology groups. In each degree n, there is a map from the homology of A intersect B in degree n to the direct sum of the homologies of A and B in degree n, then a map to the homology of X in degree n, then a connecting homomorphism to the homology of A intersect B in degree n minus 1, and so on. Exactness at each term means the image of each map equals the kernel of the next.

Example: Homology of the Two-Sphere

Decompose the two-sphere as the union of the northern hemisphere A and the southern hemisphere B, each homeomorphic to a disk and hence contractible. Their intersection is a thin equatorial band homotopy equivalent to the circle. The Mayer-Vietoris sequence in degree 2 gives: zero to H sub 2 of the two-sphere to H sub 1 of the circle to zero. Since H sub 1 of the circle is the integers, H sub 2 of the two-sphere is also the integers. In degree 1 the sequence shows H sub 1 of the two-sphere is zero, confirming the two-sphere is simply connected.

Example: Homology of the Torus

Decompose the torus as the union of two open cylinders whose overlap consists of two disjoint annuli. Applying the Mayer-Vietoris sequence and using the known homology of cylinders and annuli gives H sub 2 of the torus equal to the integers, H sub 1 of the torus equal to the direct sum of two copies of the integers, and H sub 0 of the torus equal to the integers. This confirms that the torus has one two-dimensional void and two independent one-dimensional loops.

Connecting Homomorphism

The connecting homomorphism in the Mayer-Vietoris sequence is the key nontrivial part. Given an n-cycle in X that comes from adding a chain in A and a chain in B, the connecting homomorphism extracts a cycle in A intersect B in degree n minus 1 by taking the boundary of the piece in A. This algebraic boundary operation captures how cycles in the total space are assembled from the pieces.

7. Cohomology and the Universal Coefficient Theorem

Cohomology is the dual of homology. It assigns abelian groups to topological spaces just as homology does, but with arrows reversed. The critical advantage of cohomology is that its groups carry a natural ring structure: the cup product, which encodes higher-dimensional intersection information not visible in homology alone.

Cochain Complexes and Coboundary Maps

A cochain complex is formed by taking the chain complex and applying the functor that takes homomorphisms into a coefficient group G. An n-cochain is a homomorphism from the group of n-chains to G. The coboundary map sends an n-cochain to an (n plus 1)-cochain by precomposing with the boundary map. The n-th cohomology group with coefficients in G is the group of n-cocycles (cochains killed by the coboundary) modulo the n-coboundaries (images of the previous coboundary).

Universal Coefficient Theorem

The Universal Coefficient Theorem for cohomology says that the n-th cohomology group with coefficients in an abelian group G fits into a short exact sequence. The sequence starts with the group of homomorphisms from H sub n into G, includes the n-th cohomology group, and ends with the Ext group of H sub (n minus 1) and G. The Ext term measures the extent to which the previous homology group fails to be free abelian.

Cup Product and Cohomology Ring

The cup product is a bilinear map that takes a p-cochain and a q-cochain to a (p plus q)-cochain, making the cohomology groups into a graded ring. Two spaces can have isomorphic homology and cohomology groups as abelian groups but nonisomorphic cohomology rings. The cohomology ring of the torus is the exterior algebra on two degree-1 classes, while the cohomology ring of the two-sphere wedge with the circle is different even though they have the same homology groups in every degree.

Poincaré Duality

For a closed orientable n-manifold M, Poincaré duality gives an isomorphism between the k-th cohomology group and the (n minus k)-th homology group. This duality reflects the complementary nature of k-dimensional and codimension-k submanifolds. Poincaré duality implies that the Betti numbers of a closed orientable manifold are symmetric about the middle dimension, and that the top homology group of a connected closed orientable manifold is isomorphic to the integers.

8. CW Complexes and Cellular Homology

CW complexes provide the preferred class of spaces in algebraic topology: they are flexible enough to model any space up to homotopy equivalence and rigid enough to enable explicit computation via cellular homology.

Building CW Complexes

A CW complex is built inductively. Start with a discrete set of points, the 0-skeleton. Attach 1-cells (line segments) by gluing their two endpoints to 0-cells, forming the 1-skeleton. Attach 2-cells (disks) by gluing their boundary circles continuously to the 1-skeleton. Continue attaching cells of each dimension. The attaching map for each cell is a continuous map from the boundary sphere of that cell into the existing skeleton.

Cellular Homology

For a CW complex, cellular homology computes the singular homology groups using only the cells and their attaching maps. The cellular chain group in degree n is the free abelian group on the n-cells. The cellular boundary map sends each n-cell to a linear combination of (n minus 1)-cells, with coefficients given by the degrees of the attaching maps composed with projection to each (n minus 1)-cell. Cellular homology is isomorphic to singular homology but far more computationally tractable.

Degree of a Map

The degree of a continuous map from the n-sphere to itself is the integer that gives the induced map on H sub n of the sphere. The identity map has degree 1, a reflection has degree negative 1, and the constant map has degree 0. The degree measures how many times the domain wraps around the target, counted with orientation. Degree is the key ingredient in the cellular boundary map for CW complexes.

Homology of Projective Spaces

Real projective space RP sup n has a CW structure with one cell in each dimension from 0 to n. The cellular boundary maps alternate between 0 and multiplication by 2. This gives homology groups that are the integers in degree 0, the integers mod 2 in odd degrees up to n minus 1 (if n is even), and the integers in degree n if n is odd. Complex projective space CP sup n has cells only in even dimensions, giving homology equal to the integers in each even degree from 0 to 2n and 0 elsewhere.

9. Euler Characteristic and Genus of Surfaces

The Euler characteristic is one of the oldest and most fundamental topological invariants. It unifies a variety of counting formulas and connects to the classification of compact surfaces.

Definition via Betti Numbers

The n-th Betti number of a space is the rank of its n-th homology group. The Euler characteristic is the alternating sum of the Betti numbers: subtract the first Betti number from the zeroth, add the second, subtract the third, and so on. For finite CW complexes this equals the alternating sum of the number of cells in each dimension, which is the classical Euler formula: vertices minus edges plus faces.

Classification of Compact Surfaces

Every compact connected surface without boundary is homeomorphic to exactly one of the following: the sphere, a connected sum of g tori (an orientable surface of genus g), or a connected sum of k projective planes (a non-orientable surface). Two compact surfaces are homeomorphic if and only if they have the same Euler characteristic and the same orientability. This is one of the great classification theorems of low-dimensional topology.

Gauss-Bonnet Theorem

The Gauss-Bonnet theorem connects the Euler characteristic to differential geometry: the integral of the Gaussian curvature over a closed surface equals 2 pi times the Euler characteristic. For a sphere of radius r the Gaussian curvature is 1 over r squared everywhere, and integrating over the surface area gives 4 pi, which equals 2 pi times 2, confirming Euler characteristic 2. For a flat torus the Gaussian curvature is zero, consistent with Euler characteristic 0.

10. Higher Homotopy Groups and Fibrations

Higher homotopy groups generalize the fundamental group to detect holes in all dimensions. They are more subtle and harder to compute than homology groups, reflecting deeper aspects of the global shape of a space.

Definition of Higher Homotopy Groups

The n-th homotopy group of a pointed space X, written pi sub n of X at basepoint x, is the set of homotopy classes of continuous maps from the n-sphere to X sending a basepoint on the sphere to x, with homotopies fixing the basepoint. The group operation is defined by pinching the n-sphere along an equatorial hyperplane and mapping one hemisphere via the first map and the other via the second. For n at least 2, all higher homotopy groups are abelian, unlike the fundamental group which may be nonabelian.

Whitehead's Theorem

A map between simply connected CW complexes that induces isomorphisms on all homology groups also induces isomorphisms on all homotopy groups and is therefore a homotopy equivalence. Whitehead's theorem shows that for CW complexes with trivial fundamental group, homology alone determines the homotopy type. This fails for spaces that are not CW complexes and for spaces with nontrivial fundamental group.

Fiber Bundles and Fibrations

A fiber bundle is a map p from a total space E to a base space B such that every point of B has an open neighborhood over which the bundle looks like a product of that neighborhood with a fixed fiber space F. Fiber bundles generalize covering spaces (where the fiber is discrete) and vector bundles (where the fiber is a vector space). A fibration is a weaker notion requiring only the homotopy lifting property for all spaces, not the homeomorphism condition.

Long Exact Sequence of a Fibration

Given a fibration with fiber F, total space E, and base space B, there is a long exact sequence of homotopy groups connecting pi sub n of F, pi sub n of E, pi sub n of B, pi sub (n minus 1) of F, and so on down to pi sub 0. This sequence is the main tool for computing homotopy groups. For covering spaces the long exact sequence reduces to an isomorphism between the higher homotopy groups of the total space and those of the base.

The Hopf Fibration

The Hopf fibration is a fiber bundle with total space the three-sphere, base space the two-sphere, and fiber the circle. It is defined by viewing the three-sphere as the unit sphere in the space of pairs of complex numbers and sending each point to the corresponding point of the complex projective line, which is homeomorphic to the two-sphere. The long exact sequence of the Hopf fibration gives that pi sub 3 of the two-sphere is the integers, generated by the Hopf map. This was historically surprising: the three-sphere maps nontrivially to the two-sphere even though the three-sphere is simply connected.

11. Classical Theorems and Applications

Algebraic topology produces some of mathematics' most striking theorems: results that are easy to state, hard to prove by elementary means, and yet follow cleanly from the algebraic machinery developed above.

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