Homotopy, Homology, and Geometric Invariants

No coordinate chart can detect how an entire manifold is connected, whether a closed loop can shrink, or whether a closed surface bounds a region. Homotopy and homology turn these global questions into invariants. This page develops only the tools most often needed in geometry: connected components, the fundamental group, singular homology, cohomology pairings, and Euler characteristic. They distinguish useful examples and explain global flux sectors without attempting a full algebraic-topology course.

Required background. Tensor Fields and Natural Operations supplies the smooth tensor setting used when topological invariants are paired with geometric fields; Smooth Manifolds and Coordinate Atlases and Smooth Maps, Pushforwards, and Pullbacks supply manifolds and smooth maps. Exterior Algebra and Differential Forms supplies forms and pullbacks, while de Rham Cohomology and Global Flux supplies closed forms, periods, and the cohomological pairing used below.

Helpful background. Calculus and Linear Algebra supplies quotient spaces, linear maps, and kernels; The Riemann Curvature Tensor provides local geometric data to contrast with global invariants.

Homotopy and the fundamental group

Two continuous maps $f_0,f_1:X\to Y$ are homotopic if a continuous family $H:X\times[0,1]\to Y$ joins them. A space is contractible if its identity map is homotopic to a constant. Contractible coordinate neighborhoods all look topologically simple, which is precisely why local charts cannot reveal global holes.

For a based, path-connected space $(X,x_0)$, the fundamental group $\pi_1(X,x_0)$ consists of based loops modulo endpoint-preserving homotopy. Multiplication concatenates loops. A continuous map induces a homomorphism on fundamental groups, and a homotopy equivalence induces an isomorphism. Hatcher constructs this invariant and develops its basic calculations in Hatcher 2002, Chapter 1, §§1.1–1.2, pp. 25–55.

The standard examples already discriminate important geometries:

$$\pi_1(S^1)\cong\mathbb Z, \qquad \pi_1(S^2)=0, \qquad \pi_1(T^2)\cong\mathbb Z\oplus\mathbb Z.$$

The two generators of the torus correspond to loops around its two independent circles. The sphere has no noncontractible loop, but it is not contractible; higher-dimensional information is required to see its enclosed two-cycle.

Cycles, boundaries, and homology

Singular $k$-chains are finite integer combinations of continuous maps from standard $k$-simplices into $X$. The boundary map satisfies $\partial^2=0$. Thus every boundary is a cycle, and the homology group

$$H_k(X;\mathbb Z)=\ker\partial_k/\operatorname{im}\partial_{k+1}$$

measures cycles not already boundaries. Homology forgets more than homotopy does, but it is computable and naturally pairs with differential forms. The chain construction and basic calculations are developed in Hatcher 2002, Chapter 2, §§2.1–2.2, pp. 102–159.

For connected examples,

Space	$H_0$	$H_1$	$H_2$	$\chi$
$S^1$	$\mathbb Z$	$\mathbb Z$	$0$	$0$
$S^2$	$\mathbb Z$	$0$	$\mathbb Z$	$2$
$T^2$	$\mathbb Z$	$\mathbb Z^2$	$\mathbb Z$	$0$

For a space with finite-dimensional rational homology in only finitely many degrees, the Euler characteristic is the alternating sum of the Betti numbers:

$$\chi(X)=\sum_k(-1)^k\operatorname{rank}H_k(X;\mathbb Q).$$

Compact smooth manifolds satisfy these finiteness conditions. Finite generation in each degree alone would not make an infinite alternating sum well defined. The result is a coarse invariant: $S^1$ and $T^2$ both have $\chi=0$ but are not homotopy equivalent.

Cohomology and global flux

Cohomology reverses arrows and pairs with homology. For a closed differential $k$-form $\alpha$, integration defines a linear functional on smooth $k$-cycles,

$$[C]\longmapsto\int_C\alpha.$$

If $\alpha=d\beta$, Stokes’ theorem makes the integral vanish on every cycle. A nonzero period therefore proves that a closed form is not exact, as developed in de Rham Cohomology and Global Flux.

The stronger de Rham theorem says that, for a smooth manifold $M$ without boundary, integration induces a natural isomorphism

$$H^k_{\mathrm{dR}}(M)\cong H^k(M;\mathbb R).$$

Here singular cohomology is the cohomology of real-valued linear functionals on singular chains, with differential dual to $\partial$. There is one essential bridge: continuous singular simplices need not admit pullbacks of differential forms, but smooth simplices compute the same homology. Integration over those smooth simplices is a cochain map by Stokes’ theorem. On a contractible coordinate neighborhood it induces an isomorphism by the Poincaré lemma; compatible Mayer–Vietoris sequences extend this local result to the manifold. A full proof, including the smooth-chain comparison, is given in Park, undated, Theorems 2.4 and 3.3, pp. 4–8 (PDF).

The theorem translates differential forms into real topological information; it does not recover integral or torsion information. Hatcher develops singular cohomology and its pairings in Hatcher 2002, Chapter 3, §§3.1–3.2, pp. 190–229.

On $S^1$ with angular coordinate understood locally, the one-form $d\theta$ is globally defined and closed, and its period around the circle is $2\pi$. It cannot be the differential of a globally single-valued real function. Similarly, a closed two-form with nonzero integral over $S^2$ represents a nontrivial flux class. The local equation $d\alpha=0$ alone cannot decide the global class.

What these invariants do not decide

Isomorphic homology groups do not imply homeomorphic manifolds. The fundamental group can be nonabelian, while $H_1$ records only its abelianization. Real de Rham cohomology cannot see torsion classes. None of these invariants determines a metric, curvature, causal structure, or geodesic completeness. Conversely, changing a metric without changing the underlying manifold leaves them unchanged.

The assumptions also matter. Homology depends on coefficients, and compact support or relative groups are often needed on noncompact spaces or manifolds with boundary. Those variants should be selected to match the geometric question rather than treated as interchangeable.

Exercise: three basic spaces

Use the table to compute the Euler characteristics of $S^1$, $S^2$, and $T^2$. Then explain why a closed one-form with nonzero period can exist on $S^1$ and $T^2$ but not on $S^2$.

Solution

The alternating rank sums give $\chi(S^1)=1-1=0$, $\chi(S^2)=1+1=2$, and $\chi(T^2)=1-2+1=0$. A nonzero period of a closed one-form defines a nonzero class in $H^1_{\mathrm{dR}}$. Since $H_1(S^1;\mathbb R)\cong\mathbb R$ and $H_1(T^2;\mathbb R)\cong\mathbb R^2$, such pairings exist. Since $H_1(S^2;\mathbb R)=0$, every closed one-form on $S^2$ is exact and has zero period around every loop.

References

• Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2002. Chapters 0–3, pp. 1–260.
• Park, Peter S. Proof of de Rham’s Theorem. Harvard University notes, n.d. Open PDF.