Algebraic Topology

Algebraic topology is the art of translating topological problems — questions about shape, connectivity, and deformation — into algebraic ones that can be solved by calculation. The central idea is to assign algebraic invariants (groups, rings, modules) to topological spaces in such a way that spaces which are topologically equivalent produce isomorphic algebraic objects. This machinery allows us to prove, for instance, that a sphere cannot be continuously deformed into a torus, or that there is no continuous nonvanishing vector field on a sphere of even dimension — facts that are intuitively plausible but surprisingly difficult to establish rigorously without algebra. The subject draws deeply on both general topology and abstract algebra, and the techniques it develops — homology, cohomology, homotopy groups, and spectral sequences — permeate modern mathematics from differential geometry to number theory to theoretical physics.

Singular Homology

The fundamental question that homology answers is: how many “holes” of each dimension does a topological space contain? A circle has a one-dimensional hole (the gap in the middle), a sphere has a two-dimensional hole (its interior is empty), and a torus has both. Making this precise requires considerable machinery, but the intuition is geometric throughout.

The construction begins with singular simplices. An $n$-simplex is the convex hull of $n+1$ points in general position — the 0-simplex is a point, the 1-simplex is a line segment, the 2-simplex is a triangle, the 3-simplex is a tetrahedron, and so on. The standard $n$-simplex is defined by $\Delta^n = \{(t_0, \ldots, t_n) \in \mathbb{R}^{n+1} : t_i \geq 0,\, \sum t_i = 1\}$. A singular $n$-simplex in a topological space $X$ is any continuous map $\sigma: \Delta^n \to X$ — we probe the space by mapping these standard shapes into it in all possible ways. The group $C_n(X)$, called the $n$-th chain group, is the free abelian group generated by all singular $n$-simplices. An element of $C_n(X)$ is a finite formal sum $\sum n_i \sigma_i$ with integer coefficients, called a $n$-chain.

The key algebraic operation is the boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$. For a singular $n$-simplex $\sigma$, the boundary is the alternating sum of its faces: $\partial_n \sigma = \sum_{i=0}^{n} (-1)^i \sigma \circ d_i$, where $d_i$ is the $i$-th face inclusion that omits the $i$-th vertex. The alternating signs are crucial. They ensure that boundaries cancel properly, yielding the fundamental identity $\partial_{n-1} \circ \partial_n = 0$. This says that the boundary of a boundary is zero — in geometric terms, a surface’s boundary (a closed curve) has no boundary itself. The sequence of groups and maps

$\cdots \to C_n(X) \xrightarrow{\partial_n} C_{n-1}(X) \xrightarrow{\partial_{n-1}} \cdots \to C_0(X) \to 0$

is called the singular chain complex of $X$.

The $n$-th singular homology group is defined as $H_n(X) = \ker \partial_n / \operatorname{im} \partial_{n+1}$. Elements of $\ker \partial_n$ are cycles — chains with no boundary, the algebraic stand-ins for closed curves or surfaces. Elements of $\operatorname{im} \partial_{n+1}$ are boundaries — cycles that bound a higher-dimensional region. The quotient $H_n(X)$ measures cycles that are not boundaries: these are the “genuine” $n$-dimensional holes. The identity $\partial^2 = 0$ ensures that every boundary is a cycle, so the quotient is well-defined.

The power of this construction lies in its functoriality. A continuous map $f: X \to Y$ induces group homomorphisms $f_*: H_n(X) \to H_n(Y)$ for every $n$, and these respect composition: $(g \circ f)_* = g_* \circ f_*$. This is the content of saying that $H_n$ is a functor from topological spaces to abelian groups. Crucially, two homotopic maps induce the same map on homology, so homology is a homotopy invariant: homotopy equivalent spaces have isomorphic homology groups. Computing a few standard examples anchors the theory. The $n$-sphere $S^n$ has $H_k(S^n) \cong \mathbb{Z}$ for $k = 0$ and $k = n$, and $H_k(S^n) = 0$ otherwise. A single point has $H_0(\text{pt}) \cong \mathbb{Z}$ and $H_n(\text{pt}) = 0$ for $n > 0$. The torus $T^2 = S^1 \times S^1$ has $H_0 \cong \mathbb{Z}$, $H_1 \cong \mathbb{Z}^2$, and $H_2 \cong \mathbb{Z}$, reflecting one connected component, two independent loops, and one enclosed area. These numbers record topological features that are preserved under any continuous deformation.

Cellular homology provides a far more efficient computational tool for spaces with a CW structure — spaces built by attaching disks of increasing dimension. The cellular chain complex replaces the enormous singular chain groups with small groups generated only by the cells of the CW decomposition, and the boundary map is computed from the degree of certain attaching maps. For a CW complex with finitely many cells, this reduces homology computations to linear algebra over finite matrices, making explicit calculations feasible.

Cohomology and Universal Coefficients

Cohomology is, in a precise sense, the dual of homology. Rather than mapping simplices into the space and summing them up, cohomology assigns values in an abelian group to each simplex. Formally, given a chain complex $C_\bullet(X)$ and an abelian group $G$, the cochain groups are $C^n(X; G) = \operatorname{Hom}(C_n(X), G)$ — the group of homomorphisms from $n$-chains to $G$. The coboundary operator $\delta^n: C^n(X; G) \to C^{n+1}(X; G)$ is the dual of the boundary operator, defined by $(\delta^n \varphi)(\sigma) = \varphi(\partial_{n+1} \sigma)$. Again $\delta^2 = 0$, and the $n$-th cohomology group is $H^n(X; G) = \ker \delta^n / \operatorname{im} \delta^{n-1}$.

The relationship between homology and cohomology is made precise by the Universal Coefficient Theorem. For cohomology with coefficients in an abelian group $G$, there is a natural short exact sequence:

$0 \to \operatorname{Ext}^1(H_{n-1}(X), G) \to H^n(X; G) \to \operatorname{Hom}(H_n(X), G) \to 0$

and this sequence splits (though not naturally). This theorem says that cohomology is almost determined by homology — the main term is $\operatorname{Hom}(H_n(X), G)$, which is the expected “dual” of $H_n(X)$, but there is a correction term $\operatorname{Ext}^1$ that captures torsion phenomena. For coefficient groups like $\mathbb{Q}$ or $\mathbb{R}$ where $\operatorname{Ext}^1$ vanishes, cohomology is simply the algebraic dual of homology. For $G = \mathbb{Z}$, torsion in $H_{n-1}(X)$ contributes to $H^n(X)$ through the $\operatorname{Ext}$ term, shifting dimension by one. An analogous universal coefficient theorem for homology with coefficients uses $\operatorname{Tor}$ in place of $\operatorname{Ext}$: $0 \to H_n(X) \otimes G \to H_n(X; G) \to \operatorname{Tor}(H_{n-1}(X), G) \to 0$.

What makes cohomology genuinely more powerful than homology is its ring structure. The cup product $\smile: H^p(X; R) \times H^q(X; R) \to H^{p+q}(X; R)$ is defined at the cochain level: $(\varphi \smile \psi)(\sigma) = \varphi(\sigma|_{[v_0, \ldots, v_p]}) \cdot \psi(\sigma|_{[v_p, \ldots, v_{p+q}]})$, where the vertical bars denote restriction to a front and back face. This product is associative and distributes over addition, making the direct sum $H^*(X; R) = \bigoplus_n H^n(X; R)$ into a graded ring — the cohomology ring of $X$. Moreover, the cup product is graded-commutative: $\alpha \smile \beta = (-1)^{pq} \beta \smile \alpha$ for $\alpha \in H^p$ and $\beta \in H^q$.

The cohomology ring carries strictly more information than the individual groups. A striking example: the spaces $S^2 \vee S^4$ and $\mathbb{CP}^2$ (complex projective plane) have identical homology and cohomology groups, but different cup product structures. In $\mathbb{CP}^2$, the generator $\alpha \in H^2$ satisfies $\alpha \smile \alpha \neq 0$ in $H^4$, whereas in $S^2 \vee S^4$ all cup products between positive-dimensional classes vanish. The ring structure, invisible to homology, distinguishes these spaces.

Mayer-Vietoris Theorem and Excision

Two of the most powerful computational tools in homology are the Mayer-Vietoris sequence and the excision theorem. Both are versions of the same idea: homology is a local-to-global invariant that can be assembled from the homology of smaller pieces.

The Excision Theorem states that if $Z \subset A \subset X$ with $\overline{Z} \subset \operatorname{int}(A)$, then the inclusion $(X \setminus Z, A \setminus Z) \hookrightarrow (X, A)$ induces isomorphisms on all relative homology groups: $H_n(X \setminus Z, A \setminus Z) \cong H_n(X, A)$ for all $n$. In other words, you can “excise” (cut out) a closed subset from the interior of $A$ without changing the relative homology — the topology near the excised region doesn’t contribute. This is a non-trivial statement and requires a careful argument using barycentric subdivision of simplices to refine any chain into one that avoids $Z$. Excision implies, among other things, the long exact sequence of a pair and the suspension isomorphism $\tilde{H}_n(X) \cong \tilde{H}_{n+1}(\Sigma X)$ relating the reduced homology of a space to that of its suspension.

The Mayer-Vietoris sequence gives a direct computational handle on spaces built from two pieces. Suppose $X = A \cup B$ where $A$ and $B$ are open subsets (or suitable subcomplexes). There is a long exact sequence:

$\cdots \to H_n(A \cap B) \xrightarrow{\Phi} H_n(A) \oplus H_n(B) \xrightarrow{\Psi} H_n(X) \xrightarrow{\partial} H_{n-1}(A \cap B) \to \cdots$

The map $\Phi$ sends a class in the intersection to its images in each piece (with a sign), $\Psi$ adds these contributions, and $\partial$ is the connecting homomorphism. The sequence terminates on the right with $H_0(A \cap B) \to H_0(A) \oplus H_0(B) \to H_0(X) \to 0$. This long exact sequence is a machine: given knowledge of the homology of $A$, $B$, and $A \cap B$, it constrains $H_n(X)$ completely, up to solving extension problems. An analogous Mayer-Vietoris sequence holds for cohomology, with arrows reversed.

As an illustration, decompose the circle $S^1$ into two open arcs $A$ and $B$, each homeomorphic to an open interval, with intersection $A \cap B$ consisting of two disjoint intervals. The Mayer-Vietoris sequence yields $\cdots \to H_1(A) \oplus H_1(B) \to H_1(S^1) \to H_0(A \cap B) \to H_0(A) \oplus H_0(B) \to \cdots$. Since $A$ and $B$ are contractible, their homology vanishes in degree $\geq 1$, and $A \cap B$ has two components, so $H_0(A \cap B) \cong \mathbb{Z}^2$. The sequence then forces $H_1(S^1) \cong \mathbb{Z}$, recovering the expected result. Iterated Mayer-Vietoris arguments can compute the homology of quite complex spaces, particularly surfaces and higher-dimensional manifolds.

The long exact sequences of pairs and the connecting homomorphisms that appear throughout this theory are not accidental — they are instances of a general algebraic phenomenon captured by the Snake Lemma and the Five Lemma from homological algebra. The Snake Lemma produces a long exact sequence from a short exact sequence of chain complexes, and the Five Lemma allows one to prove that a map between long exact sequences is an isomorphism if four out of five flanking maps are isomorphisms. These diagram-chasing lemmas, once abstract, become automatic tools for the working algebraic topologist.

Fiber Bundles and Homotopy Groups

While homology captures the “global” hole structure of a space, homotopy groups capture something more refined about its connectivity. The fundamental group $\pi_1(X, x_0)$ consists of homotopy classes of loops based at $x_0$, with concatenation as the group operation. The higher homotopy groups $\pi_n(X, x_0)$ for $n \geq 2$ generalize this: $\pi_n(X, x_0)$ is the set of homotopy classes of maps from the $n$-sphere $S^n$ to $X$ sending the basepoint of $S^n$ to $x_0$. For $n \geq 2$, these groups are always abelian — a consequence of the Eckmann-Hilton argument, which shows that when two group operations commute (as concatenation in two directions on $S^n$ does for $n \geq 2$), they must coincide and be commutative.

Computing homotopy groups is notoriously difficult. The homotopy groups of spheres $\pi_k(S^n)$ remain incompletely known to this day. A sampling: $\pi_1(S^1) \cong \mathbb{Z}$; $\pi_n(S^n) \cong \mathbb{Z}$ for all $n \geq 1$; $\pi_3(S^2) \cong \mathbb{Z}$ (the famous Hopf fibration, discovered by Heinz Hopf in 1931); and $\pi_4(S^2) \cong \mathbb{Z}/2$. The Hopf fibration $S^3 \to S^2$ with fiber $S^1$ demonstrates that higher homotopy groups carry information that homology misses entirely: $H_3(S^2) = 0$, yet $\pi_3(S^2) \cong \mathbb{Z}$.

Fiber bundles provide the natural framework for understanding many spaces and their homotopy theory. A fiber bundle $F \to E \xrightarrow{p} B$ consists of a total space $E$, a base space $B$, and a fiber $F$, with the projection $p$ locally a product: every point in $B$ has a neighborhood $U$ such that $p^{-1}(U) \cong U \times F$. The long exact sequence of a fibration is one of the most important tools in the subject:

$\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots \to \pi_0(F) \to \pi_0(E) \to \pi_0(B)$

This sequence relates the homotopy groups of all three spaces and can be used to compute homotopy groups by induction. For example, the Hopf fibration $S^1 \to S^3 \to S^2$ yields the segment $\pi_3(S^1) \to \pi_3(S^3) \to \pi_3(S^2) \to \pi_2(S^1) \to \pi_2(S^3)$. Since $\pi_3(S^1) = 0$, $\pi_3(S^3) \cong \mathbb{Z}$, and $\pi_2(S^1) = 0$, the sequence collapses to $0 \to \mathbb{Z} \to \pi_3(S^2) \to 0$, giving $\pi_3(S^2) \cong \mathbb{Z}$ — a clean derivation of a non-obvious fact.

The connection between homotopy and homology is mediated by two fundamental theorems. The Hurewicz theorem states that if $X$ is $(n-1)$-connected (meaning $\pi_k(X) = 0$ for $k < n$) and $n \geq 2$, then $H_k(X) = 0$ for $0 < k < n$ and there is a natural isomorphism $\pi_n(X) \cong H_n(X)$ given by the Hurewicz homomorphism (which sends a homotopy class $[f: S^n \to X]$ to $f_*(\iota_n) \in H_n(X)$, where $\iota_n$ is the fundamental class of $S^n$). This theorem provides a ladder: at the first non-trivial level, homotopy and homology coincide, giving a way to compute at least the lowest homotopy group from homology. Whitehead’s theorem complements this: a map between simply connected CW complexes that induces isomorphisms on all homotopy groups is a homotopy equivalence. Together, these results mean that for simply connected spaces, homological algebra (which is far more computable) often captures homotopy-theoretic information.

Spectral Sequences and Characteristic Classes

The most powerful computational tool in algebraic topology — and one of the most formidable objects in all of mathematics — is the spectral sequence. A spectral sequence is a sequence of “pages,” each a bigraded abelian group $E_r^{p,q}$ with a differential $d_r: E_r^{p,q} \to E_r^{p+r, q-r+1}$ satisfying $d_r^2 = 0$. The next page is the homology of the previous: $E_{r+1}^{p,q} = H(E_r^{p,q}, d_r)$. Under favorable convergence conditions, the spectral sequence “converges” to some target — the pages stabilize at $E_\infty$, and the target groups are assembled from $E_\infty$ as extensions. The conceptual payoff is that difficult problems about the target can often be solved by tracking information through successive pages, each step solving a simpler problem.

The Serre spectral sequence, developed by Jean-Pierre Serre in his 1951 thesis, is the central example. For a fibration $F \to E \to B$ with $B$ simply connected, there is a spectral sequence with $E_2^{p,q} = H^p(B; H^q(F))$ converging to $H^*(E)$. This allows the cohomology of the total space to be assembled from the cohomologies of the base and fiber. The Serre spectral sequence has been used to compute homotopy groups of spheres, to establish Bott periodicity, and to prove a host of structural results about classifying spaces and loop spaces.

Characteristic classes are cohomological invariants of vector bundles — natural assignments that associate a cohomology class on the base space to each vector bundle over it, in a way that is natural with respect to pullback. The most fundamental examples are:

• Stiefel-Whitney classes $w_i(E) \in H^i(B; \mathbb{Z}/2)$ for real vector bundles, introduced by Eduard Stiefel and Hassler Whitney in the 1930s–40s. They detect obstructions to the existence of linearly independent sections. The total Stiefel-Whitney class is $w(E) = 1 + w_1(E) + w_2(E) + \cdots \in H^*(B; \mathbb{Z}/2)$, and it satisfies the Whitney product formula $w(E \oplus F) = w(E) \smile w(F)$.
• Chern classes $c_i(E) \in H^{2i}(B; \mathbb{Z})$ for complex vector bundles, the complex analogs. They live in even-dimensional integral cohomology and satisfy analogous formulas. The Chern character $\operatorname{ch}(E) = \sum_k c_k(E)/k!$ is a ring homomorphism from K-theory to rational cohomology.
• Pontryagin classes $p_i(E) \in H^{4i}(B; \mathbb{Z})$ for real vector bundles, defined as $p_i(E) = (-1)^i c_{2i}(E_\mathbb{C})$ where $E_\mathbb{C}$ is the complexification.

Characteristic classes are powerful for several reasons. They are computable from explicit descriptions of bundles, they obstruct the existence of geometric structures (for example, $w_1(TM) = 0$ characterizes orientable manifolds), and they encode information about manifolds via Poincaré duality. For a closed oriented $n$-manifold $M$, Poincaré duality provides isomorphisms $H^k(M) \cong H_{n-k}(M)$ for every $k$, reflecting a deep symmetry between “cycles” and “cocycles.” The cup product structure, combined with Poincaré duality, defines an intersection form on the middle cohomology that is a central invariant in manifold classification.

The Chern-Weil theory, developed in the 1940s by Shiing-Shen Chern and André Weil, provides a differential-geometric construction of characteristic classes using connections and curvature on vector bundles. A connection on a bundle determines a curvature form $\Omega$, and applying an invariant polynomial to $\Omega$ yields a closed differential form whose de Rham cohomology class is independent of the connection chosen — it is the characteristic class. This deep link between topology, geometry, and analysis is a precursor to the Atiyah-Singer index theorem.

K-Theory and Cobordism

K-theory is a generalized cohomology theory built from vector bundles. Given a compact Hausdorff space $X$, the set of isomorphism classes of complex vector bundles over $X$ forms a commutative monoid under direct sum. The Grothendieck group construction (adding formal inverses) produces $K^0(X)$, or simply $K(X)$ — the zeroth complex K-group of $X$. An element of $K(X)$ is a formal difference $[E] - [F]$ of vector bundle classes, called a virtual bundle. The tensor product of bundles makes $K(X)$ into a commutative ring. Real K-theory $KO(X)$ is the analogous construction with real vector bundles.

K-theory is a cohomology theory in the generalized sense: it satisfies all the Eilenberg-Steenrod axioms except the dimension axiom. The most striking fact about complex K-theory is Bott periodicity, proved by Raoul Bott in 1959: there is a natural isomorphism $K^{n+2}(X) \cong K^n(X)$ for all $n$, where $K^n(X) = K(\Sigma^{-n} X)$ (with $\Sigma$ the suspension functor). Equivalently, $\pi_k(BU \times \mathbb{Z}) \cong \pi_{k+2}(BU \times \mathbb{Z})$, and the homotopy groups of the infinite unitary group $U = \varinjlim U(n)$ are periodic with period 2: $\pi_{2k-1}(U) \cong \mathbb{Z}$ and $\pi_{2k}(U) = 0$. For real K-theory, periodicity has period 8 — a deeper and more subtle result tied to the structure of Clifford algebras and the division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$.

The Atiyah-Hirzebruch spectral sequence connects K-theory to ordinary cohomology: $E_2^{p,q} = H^p(X; K^q(\text{pt})) \Rightarrow K^{p+q}(X)$. Since $K^q(\text{pt}) \cong \mathbb{Z}$ for $q$ even and $0$ for $q$ odd (by Bott periodicity), this simplifies to $E_2^{p,q} = H^p(X; \mathbb{Z})$ for $q$ even, converging to $K^*(X)$. The Chern character provides a rational isomorphism $\operatorname{ch}: K(X) \otimes \mathbb{Q} \xrightarrow{\sim} H^{\text{even}}(X; \mathbb{Q})$, confirming that K-theory and even-dimensional cohomology carry the same rational information, with K-theory being a finer integral invariant.

Cobordism theory studies manifolds up to the equivalence relation of bounding a manifold of one higher dimension: two closed $n$-manifolds $M$ and $N$ are cobordant if there exists a compact $(n+1)$-manifold $W$ with boundary $\partial W = M \sqcup N$. The set of cobordism classes forms an abelian group under disjoint union (with the empty manifold as the identity), and Cartesian product makes the direct sum of all cobordism groups into a graded ring — the cobordism ring. For unoriented cobordism $\mathfrak{N}_*$, a theorem of René Thom (1954, his Fields Medal work) identifies this ring with the homotopy groups of a particular spectrum: $\mathfrak{N}_n \cong \pi_n(MO)$, where $MO$ is the Thom spectrum for the orthogonal group. Thom further showed that $\mathfrak{N}_* \cong (\mathbb{Z}/2)[x_2, x_4, x_5, x_6, \ldots]$, a polynomial ring with one generator in each degree not of the form $2^k - 1$. For oriented cobordism $\Omega_*^{\text{SO}}$, the ring structure is more subtle but was later computed by Thom and others in terms of Pontryagin numbers.

Index Theory and Generalized Homology

The Atiyah-Singer index theorem, published by Michael Atiyah and Isadore Singer in 1963, is one of the great unifying theorems of twentieth-century mathematics. It equates an analytic quantity — the Fredholm index of an elliptic differential operator — with a topological quantity computed from characteristic classes. For an elliptic operator $D: \Gamma(E) \to \Gamma(F)$ on a compact manifold $M$, the index is $\operatorname{ind}(D) = \dim \ker D - \dim \operatorname{coker} D$, a finite integer. The theorem states:

$\operatorname{ind}(D) = \int_M \operatorname{ch}(\sigma(D)) \smile \operatorname{Td}(TM_\mathbb{C})$

where $\sigma(D)$ is the principal symbol of $D$ and $\operatorname{Td}$ is the Todd class. This single formula encompasses many classical results as special cases: the Gauss-Bonnet theorem (for the de Rham operator), the Hirzebruch signature theorem (for the signature operator on an oriented manifold), and the Riemann-Roch theorem (for the Dolbeault operator on a complex manifold). The index theorem demonstrates that local analytic data (the symbol of a differential operator) determines global topological invariants.

The proof of the index theorem, and subsequent developments, rely on the full machinery of algebraic topology: K-theory, characteristic classes, the Thom isomorphism, and cobordism. The Thom isomorphism theorem in K-theory states that for an oriented real vector bundle $E \to X$ of rank $r$, there is an isomorphism $K(X) \cong \tilde{K}(\operatorname{Th}(E))$ where $\operatorname{Th}(E) = D(E)/S(E)$ is the Thom space (the disk bundle quotiented by the sphere bundle). This is the K-theoretic analog of the classical Thom isomorphism in cohomology, and it is essential in the proof of Bott periodicity via the periodicity of $K(\mathbb{C}P^1) \cong \mathbb{Z}^2$.

Generalized homology theories — homology theories beyond ordinary singular homology — form the ultimate abstraction of the subject. The Eilenberg-Steenrod axioms (homotopy invariance, long exact sequence of a pair, excision, and the dimension axiom) define ordinary homology uniquely up to isomorphism. Dropping the dimension axiom (which fixes the homology of a point) opens up a vast landscape: K-theory, cobordism, stable homotopy, and elliptic cohomology are all generalized (co)homology theories. By the Brown representability theorem (1962), every generalized cohomology theory is represented by a spectrum — a sequence of spaces $\{E_n\}$ with homeomorphisms $E_n \simeq \Omega E_{n+1}$, where $\Omega$ denotes the loop space. The theory of spectra makes rigorous the idea of a “stable” homotopy type and provides a unified framework for all these generalized invariants. The stable homotopy category, built from spectra with the smash product as a symmetric monoidal structure, is one of the most active arenas in contemporary algebraic topology, with ongoing work on chromatic homotopy theory, topological modular forms, and motivic homotopy theory connecting algebraic topology to algebraic geometry and number theory.

Invariant	Type	Captures	Example computation
Singular homology $H_n(X; \mathbb{Z})$	Abelian groups	Holes by dimension	$H_n(S^n) \cong \mathbb{Z}$, $H_k(S^n) = 0$ for $0 < k \neq n$
Cohomology ring $H^*(X; R)$	Graded ring	Holes + cup product	$H^*(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}[\alpha]/(\alpha^{n+1})$
Homotopy groups $\pi_n(X)$	Groups (abelian for $n \geq 2$)	Maps from spheres	$\pi_3(S^2) \cong \mathbb{Z}$ (Hopf)
K-theory $K(X)$	Commutative ring	Vector bundle classes	$K(S^{2n}) \cong \mathbb{Z}^2$ (Bott periodicity)
Cobordism $\Omega_n^{\text{SO}}$	Abelian group	Manifold bordism	$\Omega_4^{\text{SO}} \cong \mathbb{Z}$ (signature)