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Sign local system on real projective space

Statement

For $n = 1$ , identify $π_{1} ({R P}^{1}) ≅ Z$ and let $m \in Z$ act on $Z_{s g n}$ as multiplication by $(- 1)^{m}$ . For $n \geq 2$ , let the unique nonidentity element $g \in π_{1} ({R P}^{n}) ≅ Z / 2$ act as $- 1$ . With one cell in every degree $0 \leq k \leq n$ , the cellular differential is zero for even $k$ and multiplication by $2$ up to a harmless sign for odd $k$ . Thus $H_{k} ({R P}^{n}; Z_{s g n}) ≅ {\begin{cases} Z / 2, & k = 0, \\ Z / 2, & 0 < k < n and k is even, \\ 0, & 0 < k < n and k is odd, \\ Z, & k = n and n is even, \\ 0, & k = n and n is odd, \end{cases}$ and it is zero outside $0 \leq k \leq n$ . In particular the top group is $Z$ exactly when $n$ is even; in that case the sign system is the orientation system.

Facts & Assumptions

Given: $n \geq 1$ , the standard projective CW structure, and the sign system.

[F1]

Cellular chains compute local homology evaluates lifted group-ring incidence matrices through monodromy.

[F2]

The orientation system is a local system identifies orientation monodromy with the orientation character.

[F3]

The degree map identifies the fundamental group of the circle with $Z$ , sending the positive once-around loop to $1$ ( $Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism).

[F4]

The antipodal self-map of $S^{n}$ , for $n \geq 1$ , has degree $(- 1)^{n + 1}$ (Degree of identity constant reflection and antipodal sphere maps).

[F5]

For every $n \geq 2$ , the sphere $S^{n}$ is simply connected ( $S^{n}$ is simply connected for every $n \geq 2$ ).

[F6]

The deck group of a universal cover of a connected, locally path-connected, semilocally simply connected base is its fundamental group (For a path-connected locally path-connected semilocally simply connected base, the deck group of a universal cover is isomorphic to the fundamental group).

Proof

technique · direct

1.1

Compute the cellular differential, including $n = 1$ . [F1, F3, F5, F6] For $n = 1$ , the map $t \mapsto [\cos (π t) : \sin (π t)]$ identifies $R / Z$ with ${R P}^{1}$ . Under [F3], the positive once-around loop is a generator $g$ of its infinite cyclic fundamental group. Choose the vertex lift at $0 \in R$ and the lifted open edge from $0$ to $1$ . Its boundary is

$\partial {\tilde{e}}_{1} = g \tilde{v} - \tilde{v} .$

Evaluation through the action $g \mapsto - 1$ gives $- 2$ . Choosing the opposite edge or vertex-lift convention gives $g^{- 1} - 1$ , which also evaluates to $- 2$ ; reversing its orientation changes this to $2$ . This is a direct universal-cover calculation on $R$ , not an assertion that $S^{1} \to {R P}^{1}$ is universal.

For $n \geq 2$ , the antipodal quotient map $S^{n} \to {R P}^{n}$ is a two-sheeted cover; [F5] makes it the universal cover. Projective coordinate charts make the connected base locally path-connected and semilocally simply connected, so [F6] identifies its fundamental group with the deck group $⟨ g ∣ g^{2} = 1 ⟩$ . In the lifted standard projective CW structure, the two hemispherical faces of a lifted $k$ -cell contribute $1$ and $(- 1)^{k} g$ : the antipodal gluing preserves the induced face orientation for even $k$ and reverses it for odd $k$ . Thus the group-ring boundary is $1 + (- 1)^{k} g$ . Evaluating at $g = - 1$ gives $1 - (- 1)^{k}$ , hence zero for even $k$ and $2$ for odd $k$ . Together with the direct $n = 1$ calculation, [F1] gives the asserted differential in every allowed dimension. Reversing a cell orientation changes only its harmless overall sign.

2.1

The resulting complex has one copy of $Z$ in each degree. For $0 < k < n$ , an even $k$ has zero outgoing differential and incoming image $2 Z$ , giving $Z / 2$ ; an odd $k$ has injective outgoing differential, giving zero. At degree zero, $d_{1} = 2$ gives $Z / 2$ . At the top there is no incoming differential, so the kernel is $Z$ for even $n$ and zero for odd $n$ . This proves the table.

step 1.1

3.1

Compare with the orientation system in both ranges. [F2, F4, step 1.1, step 2.1] When $n = 1$ , the displayed identification with $R / Z$ gives the projective line its usual circle orientation. Its orientation character is therefore trivial, whereas the positive generator acts by $- 1$ on $Z_{s g n}$ . Hence the sign system is not the orientation system, consistently with the zero top sign homology in step 2.1.

For $n \geq 2$ , the deck transformation of the universal sphere cover is antipodal and has degree $(- 1)^{n + 1}$ by [F4]. It reverses local orientation exactly when $n$ is even. By [F2], its orientation monodromy is therefore $- 1$ exactly for even $n$ , so $Z_{s g n}$ equals $O_{{R P}^{n}}$ exactly in that case; the top $Z$ in step 2.1 is then its twisted fundamental class. This proves both directions of “exactly when”: $n = 1$ was separated, odd $n \geq 3$ has trivial orientation monodromy but nontrivial sign monodromy, and even $n$ has the same nontrivial monodromy in both systems.

For $n = 0$ , outside the stated range, ${R P}^{0}$ is a point with trivial fundamental group, so no nontrivial sign system exists and its ordinary $H_{0}$ is $Z$ . All lifts and orientations above are individually specified finite data, so no AC is used. ∎

Depends on

Used by

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Sources

Davis and Kirk, Lecture Notes in Algebraic Topology, Chapter 5 §2.1, Exercise 77, pp.99–100 (standard reference, not scraped)