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Wu classes of a closed surface

Example

Assume AC, and let $M$ be a nonempty closed connected topological surface; thus $M$ is compact, boundaryless, and two-dimensional. Choose a generator $e_{x}$ of the integral orientation stalk $O_{x}$ at every $x \in M$ . For a singular one-simplex $σ$ from $v_{0}$ to $v_{1}$ , define $ϵ (σ) \in F_{2}$ by

T_{σ} (e_{v_{0}}) = (- 1)^{ϵ (σ)} e_{v_{1}} .

The verification below proves that $ϵ$ is a cocycle and that its class is independent of the chosen generators. Write $w_{1} (M) = [ϵ] \in H^{1} (M; F_{2})$ . Then the Wu classes of $M$ are

v_{0} (M) = 1, v_{1} (M) = w_{1} (M), v_{i} (M) = 0 (i > 1) .

Moreover, $v_{1} (M) = 0$ exactly when $M$ is orientable.

Facts & Assumptions

Given: The surface $M$ , the family $(e_{x})_{x \in M}$ , and the cochain $ϵ$ specified above.

[F1]

Orientation local system and orientation cover defines the infinite cyclic stalks $O_{x}$ , their path transport, and the two-sheeted orientation cover. Transport is unchanged by endpoint-fixed homotopy and respects path concatenation.

[F2]

The declared supplier prop-the-manifold-orientation-system-is-a-local-system regards $O_{M}$ as a covariant integral local system and identifies a continuous generator section with an orientation.

[F3]

The declared supplier def-singular-and-cellular-chain-complexes-with-local-coefficients places a local coefficient at the first vertex and, on a one-simplex, gives

(δ c) (σ) = T_{σ}^{- 1} (c (v_{1})) - c (v_{0}) .

The same definition gives local chains as direct sums, hence as finite chains. The local differentials square to zero by the declared supplier lem-twisted-boundaries-square-to-zero-and-are-independent-of-lift-bases.

[F4]

The declared supplier lem-canonical-twisted-fundamental-classes-over-compact-subsets gives the canonical class $[M]^{t w} \in H_{2} (M; O_{M})$ whose local value at $x$ is $o_{x} \otimes o_{x}$ , independently of the sign of the generator $o_{x}$ .

[F5]

The declared supplier def-cup-and-cap-products-with-local-coefficient-pairings defines the cohomology-first local cap product and fixes its chain sign:

\partial (ϕ \cap c) = (- 1)^{p} (ϕ \cap \partial c - (δ ϕ) \cap c) (ϕ \in C^{p}) .

[F6]

Fundamental class of a compact oriented manifold characterizes the ordinary mod-two fundamental class by its nonzero value in every local top-homology stalk.

[F7]

Bockstein connecting operation defines the mod-two Bockstein from $0 \to F_{2} \overset{2}{\to} Z / 4 \to F_{2} \to 0$ by lifting a cocycle and dividing its coboundary. The canonical zero/one lift is choice-free. Sq^1 is the mod-two Bockstein identifies this operation with $S q^{1}$ .

[F8]

Singular cochain complex with coefficients uses the positive ordinary coboundary convention $(δ a) (z) = a (\partial z)$ .

[F9]

Steenrod normalization, instability, suspension, and top square gives $S q^{0} x = x$ , $S q^{k} x = 0$ for $k > \deg x$ , and $S q^{\deg x} (x) = x ⌣ x$ .

[F10]

Wu classes of a closed manifold defines $v_{i} (M)$ as the unique class representing the $S q^{i}$ functional under the mod-two cup pairing, and sets it to zero outside $0 \leq i \leq \dim M$ .

[A1]

The Axiom of Choice is used directly once: from the nonempty two-element set of generators of every stalk $O_{x}$ , it supplies the set-indexed family $(e_{x})_{x \in M}$ . It is also inherited through [F10]'s perfect-pairing result. No later family of representatives, paths, charts, or primitives is chosen.

Verification

Proof technique: compare the mod-four cohomology Bockstein pairing with the integral homology lift-and-divide cycle obtained from the twisted fundamental cycle.

1.1

The edge signs form a cocycle. [F1, F2, A1, given] For a singular two-simplex, write $ϵ_{i j}$ for the sign on its affine edge from vertex $i$ to vertex $j$ . The $02$ edge is homotopic relative to its endpoints to the $01$ edge followed by the $12$ edge. Functoriality of orientation transport therefore gives

ϵ_{02} = ϵ_{01} + ϵ_{12} in F_{2} .

With the positive singular coboundary, $(δ ϵ) (012) = ϵ_{12} - ϵ_{02} + ϵ_{01} = 0$ . Hence $ϵ$ is a cocycle. A degenerate edge has identity transport and therefore sign zero, consistently with this calculation.

2.1

The cohomology class does not depend on the generator family. [F1, step 1.1] Any other family has the form $e_{x}^{'} = (- 1)^{t (x)} e_{x}$ for a unique ordinary zero-cochain $t : M \to F_{2}$ . Its edge signs satisfy

ϵ^{'} (σ) = ϵ (σ) + t (v_{1}) - t (v_{0}) = ϵ (σ) + (δ t) (σ) .

Thus $[ϵ^{'}] = [ϵ]$ , so $w_{1} (M)$ is well defined.

3.1

The signed generator cochain has an even coboundary whose half reduces to $ϵ$ . [F1, F3, step 1.1, step 2.1] Define the local zero-cochain $c \in C^{0} (M; O_{M})$ by $c (x) = e_{x}$ . Since $T_{σ} (e_{v_{0}}) = (- 1)^{ϵ (σ)} e_{v_{1}}$ , the formula in [F3] gives

(δ c) (σ) = ((- 1)^{ϵ (σ)} - 1) e_{v_{0}} .

Consequently there is a unique local one-cochain $b$ with $δ c = 2 b$ : $b (σ) = 0$ when $ϵ (σ) = 0$ and $b (σ) = - e_{v_{0}}$ when $ϵ (σ) = 1$ . Since local cochain groups are products of infinite cyclic groups, they have no two-torsion. Thus $2 δ b = δ^{2} c = 0$ implies $δ b = 0$ .

There is a canonical morphism of local systems $r : O_{M} \to {\underline{F}}_{2}$ : if $e$ is either generator, $r (n e) = n m o d 2$ . The formula is independent of replacing $e$ by $- e$ , and orientation transport changes a generator only by sign, so it commutes with transport. The displayed values of $b$ give $r (b) = ϵ$ .

4.1

Cap the twisted fundamental cycle with the generator cochain. [F3, F4, F5, F6, step 3.1] There is a canonical local-coefficient pairing

q : O_{M} \otimes O_{M} ⟶ \underline{Z}, q_{x} (n e, m e) = n m,

where $e$ is either generator of $O_{x}$ . Simultaneously replacing $e$ by $- e$ leaves $n m$ unchanged, and simultaneous orientation transport does the same, so $q$ is well defined and transport-compatible.

Choose one finite twisted cycle $C$ representing $[M]^{t w}$ ; this is a single existential witness, not a family of choices. Applying $r$ to its coefficients gives an ordinary mod-two cycle $\overline{C}$ . At every point, the canonical local value $o_{x} \otimes o_{x}$ from [F4] maps to the unique nonzero mod-two local orientation. The uniqueness in [F6] therefore gives $[\overline{C}] = [M]_{2}$ .

Put $Z = c \cap_{q} C \in C_{2} (M; Z)$ . On each simplex, reduction modulo two turns $q$ into multiplication in $F_{2}$ , turns $r (c)$ into the constant zero-cochain $1$ , and turns $C$ into $\overline{C}$ . Hence

\overline{Z} = 1 \cap \overline{C} = \overline{C} .

Thus $Z$ is an integral lift of the mod-two fundamental cycle.

5.1

The cap-boundary sign produces the correct lift-and-divide cycle. [F3, F5, step 3.1, step 4.1] Since $C$ is a cycle and $c$ has degree zero, the exact convention in [F5] gives

\partial Z = c \cap_{q} \partial C - (δ c) \cap_{q} C = - 2 (b \cap_{q} C) .

Set $W = - b \cap_{q} C$ . Then $\partial Z = 2 W$ . The ordinary singular chain group is free abelian on the singular simplices, so $2 \partial W = \partial^{2} Z = 0$ implies $\partial W = 0$ . After reducing modulo two, the minus sign disappears and step 3.1 gives

\overline{W} = ϵ \cap \overline{C} .

6.1

The mod-four $S q^{1}$ pairing is evaluation on $\overline{W}$ . [F7, F8, step 4.1, step 5.1] Let $x \in H^{1} (M; F_{2})$ , represent it by a cocycle $a$ , and let $\hat{a}$ be its canonical integer zero/one lift. There is a unique integer two-cochain $h$ such that $δ \hat{a} = 2 h$ . It is a cocycle because integer cochains have no two-torsion. Reducing $\hat{a}$ modulo four shows from [F7] that

S q^{1} (x) = [h m o d 2] .

The positive coboundary convention and $\partial Z = 2 W$ give the exact integer calculation

2 h (Z) = (δ \hat{a}) (Z) = \hat{a} (\partial Z) = 2 \hat{a} (W) .

Canceling $2$ in $Z$ and then reducing modulo two yields

⟨ S q^{1} (x), [M]_{2} ⟩ = ⟨ x, [\overline{W}] ⟩ .

7.1

Cap-cup adjunction identifies the orientation class. [F5, step 3.1, step 4.1, step 5.1, step 6.1] For the cohomology-first cap convention, evaluating $a$ on $ϵ \cap \overline{C}$ is exactly the Alexander--Whitney evaluation of $ϵ ⌣ a$ on $\overline{C}$ : $ϵ$ reads the front edge and $a$ reads the retained back edge. Therefore

\begin{aligned} ⟨ S q^{1} (x), [M]_{2} ⟩ & = ⟨ x, w_{1} (M) \cap [M]_{2} ⟩ \\ = ⟨ w_{1} (M) ⌣ x, [M]_{2} ⟩ . \end{aligned}

By [F9], this also states the surface self-intersection identity $⟨ x ⌣ x, [M]_{2} ⟩ = ⟨ w_{1} (M) ⌣ x, [M]_{2} ⟩$ ; it was derived from the chain calculation, not assumed as Wu's formula.

8.1

The degree-one Wu class is $w_{1} (M)$ . [F10, step 7.1] The identity in step 7.1 holds for every $x \in H^{1} (M; F_{2}) = H^{2 - 1} (M; F_{2})$ . By the defining uniqueness of the degree-one Wu class in [F10], it follows that $v_{1} (M) = w_{1} (M)$ .

9.1

The remaining Wu classes have the asserted values. [F9, F10, step 8.1] For $i = 0$ , [F9] makes the defining functional $x \mapsto ⟨ S q^{0} x, [M]_{2} ⟩$ equal to evaluation on the fundamental class, which is represented by the unit; uniqueness in [F10] gives $v_{0} (M) = 1$ . For $i = 2$ , the test classes in [F10] have degree zero, so [F9] gives $S q^{2} x = 0$ for all of them. The zero class represents this zero functional, and uniqueness gives $v_{2} (M) = 0$ . Indices $i > 2$ are zero by the out-of-range convention in [F10]. Hence $v_{i} (M) = 0$ for every $i > 1$ .

10.1

Vanishing of $w_{1} (M)$ is equivalent to orientability. [F1, F2, step 2.1, step 9.1] If $M$ is oriented, let $s_{x}$ be its continuous generator section. Write $s_{x} = (- 1)^{t (x)} e_{x}$ . Transport preserves $s$ , so the generator-change calculation in step 2.1 gives $ϵ = δ t$ and hence $w_{1} (M) = 0$ .

Conversely, if $w_{1} (M) = 0$ , choose an ordinary zero-cochain $t$ with $ϵ = δ t$ and set $e_{x}^{'} = (- 1)^{t (x)} e_{x}$ . Step 2.1 shows that all edge signs for $(e_{x}^{'})$ vanish. Thus transport along every singular path carries its initial $e^{'}$ to its terminal $e^{'}$ . Around any point, take a path-connected orientation-chart ball and the basic local orientation section whose value at that point is $e^{'}$ . Transport inside the ball generates that section, so the path-transport property makes it equal to $e^{'}$ throughout the ball. Hence $x \mapsto e_{x}^{'}$ is locally continuous, and therefore is a global section of the orientation cover. By [F2], it orients $M$ . Since $v_{1} = w_{1}$ by step 8.1, this proves the final biconditional.

11.1

Boundary and choice cases are explicit. [F3, F4, F5, F7, F8, F10, A1, step 1.1, step 3.1, step 4.1, step 5.1, step 6.1, step 7.1, step 8.1, step 9.1, step 10.1] The hypothesis excludes the empty and disconnected cases and fixes dimension two; closed excludes manifold boundary. Zero classes $x$ are included in step 6.1. The degree endpoints $i = 0, 1, 2$ and all out-of-range indices were handled in step 9.1. Degenerate simplices remain in the unnormalized singular complexes; their ordinary and local boundary formulas are the ones used above. The two divisions by $2$ are unique because the relevant integral cochain and chain groups are torsion-free. The zero/one lift of $a$ , the reductions $r$ , and the pairing $q$ are canonical. Apart from the one pointwise generator-family selection declared in [A1], only the single cycle representative $C$ , the single primitive $t$ under the hypothesis $w_{1} = 0$ , and one chart at a time are chosen; these are ordinary existential instantiations, not further uses of AC. ∎

Remarks

The five suppliers used in [F2]--[F5] are homed on the later page local-coefficients-twisted-homology-and-duality (batch 5): this examples page precedes that page in the reading order, and the batch-5 manifest already whitelists the target page under the examples page's forwardRefs. All five items are declared in deps here, so the dependency graph is complete; because their page is later, they are named by ID in [F2]--[F5] rather than linked, since a body hyperlink to later material must be declared as a forward reference and Step-5b resolution removes that declaration. Rehoming this example to local-coefficients-twisted-homology-and-duality-examples (an owner-only reading-order change) would make every citation backward and restore the links.

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Sources

Ranicki, Algebraic and Geometric Surgery (standard reference, not scraped)
Hatcher, correction to the last two paragraphs of Algebraic Topology page 335 (standard reference, not scraped)
Davis and Kirk, Lecture Notes in Algebraic Topology (standard reference, not scraped)