For p(z)=z⁵, start at the root exp(2πi·2/5) above 1 and lift a loop making three positive turns downstairs. The lifted endpoint has sheet label 2+3 modulo 5, which is 0. The base loop closes, but this lift does not return to its own starting point. Calculate what happens after five turns instead.
A loop therefore acts as a permutation of the fiber. A deck transformation is different: it is a homeomorphism h:E→E satisfying p∘h=p on the entire covering space. We will relate these notions without identifying them automatically.
Predict
Can a deck transformation of a path-connected cover fix one point and move another? Follow a path from the fixed point to the proposed moving point. Both that path and its transformed path lift the same base path with the same start. Predict the consequence of uniqueness of path lifting.
Experiment
Write down the action on the five-sheet fiber as modular addition.
Starting sheet
Winding
Ending sheet
2
3
0
2
−1
1
2
5
2
For this cover, rotations z↦exp(2πik/5)z are deck transformations. Check directly that raising the rotated point to the fifth power gives z⁵. Explain why the same formula with an arbitrary rotation angle usually fails.
Understand
Assume B is path connected and locally path connected and the covering E is path connected. Put G=π₁(B,b₀). For a fiber point e and loop class g, define e·g as the endpoint of its lift from e. Homotopy lifting makes this independent of the representative. Because paths multiply in travel order, (e·g)·k=e·(gk): this is a right action. Reversing a loop supplies the inverse permutation.
Choose e₀ and let H be its stabilizer. A loop fixes e₀ exactly when it is the projection of a loop upstairs, so H is the subgroup induced by π₁(E,e₀). Any fiber point is reachable by a path from e₀ in E; its projected loop proves the action is transitive. The fiber identifies with right cosets H\G: H g corresponds to e₀·g, and right multiplication sends H g to H gk. To check this convention, e₀·g=e₀·k exactly when gk⁻¹ belongs to H.
A deck transformation commutes with this action by uniqueness of lifted paths. It is determined by its value at e₀, since every point can be reached by a path. A map commuting with the action and sending H to H n must send H g to H ng. This formula is a bijective well-defined map precisely when n normalizes H, meaning nHn⁻¹=H. The possible deck transformations consequently form N(H)/H. Here composition has its usual function order: the map for n composed with the map for m sends H g to H nmg, consistent with the quotient multiplication.
To realize the fiber formula on E, apply the lifting criterion to p:E→B with the new chosen starting point. Normalization gives equality of the relevant based subgroups. Lifts in the two directions compose to the identity by uniqueness. This explains why a purely algebraic fiber permutation extends to a deck homeomorphism; an arbitrary permutation need not.
The cover is regular when its deck group acts transitively on a fiber. The description above makes this equivalent to H being normal in G, in which case the deck group is G/H. The five-fold circle example is regular because 5Z is normal in Z. In a nonnormal cover, the number of deck transformations can be smaller than the number of sheets.
Apply
Let G=S₃ and H be generated by the transposition (12). Its index is three, but its normalizer is H itself: conjugation carries (12) to the transposition of the transported pair, and preserving that pair in S₃ fixes the remaining symbol. Thus any connected cover realizing this subgroup has three sheets and trivial deck group. Explain how this refutes “every sheet permutation is a deck transformation.” Existence of such a cover requires the base-space hypotheses in the next lesson.
Continue
Keep the action side and coset convention visible in every calculation. Classification will explain which subgroups occur as connected coverings. The normalizer result is developed in Hatcher, Section 1.3; the argument here tracks the travel-order convention explicitly.
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For p(z)=z⁷, start at sheet 5 and lift winding −3. Give the endpoint sheet in {0,…,6}.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
How many deck transformations does p(z)=z⁶ have as a connected circle cover?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove that lifting loop endpoints gives a right action under travel-order path multiplication. Identify the stabilizer of a fiber point e₀.
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Prove a deck transformation fixing one point of a path-connected cover is the identity. Give a disconnected counterexample to this conclusion.
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Assume the base is path connected and locally path connected and the covering is path connected. Identify the fiber with right cosets Hg and derive Tn(Hg)=Hng for n in N(H). Explain the composition order and why arbitrary fiber permutations need not extend.
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Assume the base is path connected and locally path connected and the covering is path connected. Suppose a connected cover has G=S₃ and H generated by (23). Compute its sheet number and deck group and decide regularity. State any realization qualification.
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Reflect and continue
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