izak

cohomology ring of n-sphere

Differential forms and holes in a space are related through cohomology, but the direct measure of holes is homology. In simplicial homology, a space is decomposed into simplices, and chain groups \(C_k(X)\) are formal linear combinations of \(k\)-simplices. The boundary operator \(\partial\) maps a simplex to its boundary, an alternating sum of its faces, satisfying \(\partial^2 = 0\) due to sign cancellations. Cycles (elements killed by \(\partial\)) represent loops or higher-dimensional analogs, while boundaries (images of \(\partial\)) are loops that bound filled regions. Homology \(H_k(X)\) measures holes by quotienting cycles by boundaries, capturing loops that aren’t filled in. Cohomology, on the other hand, dualizes this construction. Instead of chains, cohomology uses cochains \(C^k = \text{Hom}(C_k, \mathbb{Z})\), with a coboundary operator increasing dimension. Intuitively, cohomology measures how forms can detect holes. For instance, a closed form (cocycle) that isn’t exact (coboundary) corresponds to a hole. This duality is formalized in theorems like Poincaré duality, which relates \(k\)-chains to \((n-k)\)-cochains via intersection pairing, and the Universal Coefficient Theorem, which connects homology and cohomology over different coefficient rings. In de Rham cohomology, differential forms are used as cochains, with closed forms representing cocycles and exact forms representing coboundaries. This theory detects holes similarly to simplicial cohomology but uses smooth structures. For example, the space \(\Omega_n(X)\) of differential \(n\)-forms on \(X\) has that closed forms in \(\Omega_n(X)\) detect holes via Stokes' theorem, which ensures that integrals of closed forms over cycles are invariant under homotopy. If two paths yield different integrals for a closed form, they cannot be deformed into each other, indicating a "hole." This idea is formalized by the de Rham map, \(\Omega_n(X) \to \text{Hom}(C_n(X), \mathbb{R})\), sending a form \(\omega\) to the functional \(\sigma \mapsto \int_\sigma \omega\). Stokes' theorem ensures this is a chain map, and the de Rham theorem shows it induces an isomorphism between de Rham cohomology and singular cohomology, linking differential forms to the intuitive notion of holes. However, coefficients matter: de Rham cohomology with \(\mathbb{R}\) coefficients misses torsion phenomena, like in \(\mathbb{R}P^2\), where a loop traversed twice bounds a disk, but not once. Thus, while differential forms and holes are linked through cohomology, the choice of coefficients and theory determines what geometric or topological features are visible, giving the need for the Universal Coefficient Theorem.

The above is based on this MSE post

Every continuous map \(f: X \to Y\) induces a homomorphism from the cohomology ring of \(Y\) to that of \(X\), limiting the possible maps between spaces. Unlike more complex invariants like homotopy groups, singular cohomology is often computable for spaces of interest. The construction begins with the singular chain complex, a sequence of free abelian groups \(C_i\) generated by continuous maps from the standard \(i\)-simplex to \(X\), connected by boundary homomorphisms \(\partial_i\). The singular homology of \(X\) is the homology of this complex. To define cohomology, one fixes an abelian group \(A\) and "dualizes the chain complex, replacing \(C_i\) with \(C_i^* = \text{Hom}(C_i, A)\)"...

... e.g. consider the chain complex \(C_*\) with \(C_0 = \mathbb{Z}\), \(C_1 = \mathbb{Z}\), and \(C_i = 0\) for \(i \neq 0, 1\), connected by the boundary map \(\partial_1: C_1 \to C_0\) defined by \(\partial_1(n) = 2n\). To dualize this complex, replace each \(C_i\) with \(C_i^* = \text{Hom}(C_i, A)\), where \(A\) is an abelian group, say \(A = \mathbb{Z}\). Here, \(C_0^* = \text{Hom}(\mathbb{Z}, \mathbb{Z}) \simeq \mathbb{Z}\) and \(C_1^* = \text{Hom}(\mathbb{Z}, \mathbb{Z}) \simeq \mathbb{Z}\), with the dual map \(d_0: C_0^* \to C_1^*\) defined by \(d_0(f)(m) = f(\partial_1(m)) = f(2m)\). This dual map \(d_0\) effectively "reverses the arrows" of the original boundary map...

...and replacing \(\partial_i\) with its dual \(d_{i-1}: C_{i-1}^* \to C_i^*\). This reversal of arrows yields a cochain complex, and the \(i\)-th cohomology group \(H^i(X, A)\) is defined as \(\text{ker}(d_i) / \text{im}(d_{i-1})\). Elements of \(C_i^*\) are called \(i\)-cochains, while those in \(\text{ker}(d_i)\) and \(\text{im}(d_{i-1})\) are cocycles and coboundaries, respectively. The cohomology classes, represented by equivalence classes of cocycles, form the cohomology groups, which vanish for negative \(i\).

In singular cohomology, the cup product defines a multiplication on the cohomology ring \(H^*(X)\) of a topological space \(X\). It is constructed by defining a product on cochains: if \(\alpha^p\) is a \(p\)-cochain and \(\beta^q\) is a \(q\)-cochain, their cup product \(\alpha^p \smile \beta^q\) is a \((p+q)\)-cochain given by \((\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_{0,1,...,p}) \cdot \beta^q(\sigma \circ \iota_{p,p+1,...,p+q})\), where \(\sigma\) is a singular \((p+q)\)-simplex and \(\iota_S\) embeds the simplex spanned by \(S\) into the \((p+q)\)-simplex. Informally, \(\sigma \circ \iota_{0,1,...,p}\) represents the front \(p\)-face of \(\sigma\), while \(\sigma \circ \iota_{p,p+1,...,p+q}\) represents the back \(q\)-face. The coboundary of the cup product satisfies \(\delta(\alpha^p \smile \beta^q) = \delta\alpha^p \smile \beta^q + (-1)^p (\alpha^p \smile \delta\beta^q)\). This ensures that the cup product of two cocycles is a cocycle, and the product of a coboundary with a cocycle is a coboundary. The cup product thus induces a bilinear operation on cohomology, \(H^p(X) \times H^q(X) \to H^{p+q}(X)\), making \(H^*(X)\) a graded ring.

For how the cup product is dual to intersection for oriented minfolds, I recommend this paper by Michael Hutchings

For the \(n\)-sphere \(S^n\), we know that the only non-zero cohomology groups are \(H^0(S^n, \mathbb{Z}) \simeq \mathbb{Z}\) and \(H^n(S^n, \mathbb{Z}) \simeq \mathbb{Z}\)...

...For the \(n\)-sphere \(S^n\), the only non-zero cohomology groups are \(H^0(S^n, \mathbb{Z}) \simeq \mathbb{Z}\) and \(H^n(S^n, \mathbb{Z}) \simeq \mathbb{Z}\) because \(S^n\) is a connected (it is NOT the union of two disjoint non-empty open sets), compact (every open cover has a finite subcover), orientable manifold. Now, the $n$-sphere is a closed oriented manifold so that we can use the Poincaré duality saying the $k$-cohomology group is isomorphic to the $n-k$-th homology group: $H^k (\mathbb T^2) \cong H_{n-k} (\mathbb T^2)$. The group \(H^0(S^n, \mathbb{Z})\) corresponds to the constant functions on \(S^n\), reflecting its connectedness as there is only one such function up to scaling,. And \(H^n(S^n, \mathbb{Z})\) captures the top-dimensional cohomology, representing the orientation class of \(S^n\), e.g. (thanks to this and that MSE) for de Rham $H^n_{dR}(S^n)\simeq \mathbb{R}$ by the map $[\omega]\mapsto \int_{S^n} \omega$ where there is $\eta$ a nowhere zero $n$-form on $S$ and we can say $C = \int_S \eta \in \mathbb R$. So for any $[\omega] \in H^n(S)$, see $D:= \int_S \omega$. This is nonzero. Then $[\frac{D}{C} \eta] = [\omega]$ as $$\int_S \frac DC \eta = D = \int_S \omega . $$ And $\frac DC \eta$ is nowhere vanishing (thus an orientation). All other cohomology groups vanish because \(S^n\) has no "holes" in intermediate dimensions. ...

... The generator of \(H^0\) corresponds to the unit element of the ring, denoted as \(1\), since the cup product with \(H^0\) acts as the identity map on \(H^k(X; \mathbb{Z})\). The generator of \(H^n\) is denoted as \(x\). The cup product structure is determined by the relations \(1 \smile 1 = 1\), \(1 \smile x = x\), \(x \smile 1 = x\), and \(x \smile x = 0\), where the last relation arises because \(H^{2n}(S^n, \mathbb{Z}) = 0\). The cohomology ring \(H^*(S^n; \mathbb{Z})\) is therefore the direct sum \(H^0(S^n, \mathbb{Z}) \oplus H^n(S^n, \mathbb{Z})\), which can be expressed as \(\alpha_1 \cdot 1 \oplus \alpha_2 \cdot x\) for integers \(\alpha_1, \alpha_2\). This ring is abstractly isomorphic to the polynomial ring \(\mathbb{Z}[x]/(x^2)\), where \(x\) represents the generator of \(H^n(S^n, \mathbb{Z})\). A similar analysis applies to the real projective plane \(\mathbb{R} P^2\), whose cohomology ring is \(\mathbb{Z}[x]/(2x, x^2)\), with \(x\) being the generator of \(H^2(\mathbb{R} P^2, \mathbb{Z})\). Here, the relations \(2x = 0\) and \(x^2 = 0\) reflect the torsion in the cohomology groups and the vanishing of the cup product in higher degrees, respectively.

The above note expands on this MSE post.