Reduced homology of double loop spaces and the first nonvanishing higher homotopy group #
The reduced homology H~_*(Ω²X) of a double loop space is the kernel of the map
H_*(Ω²X) → H_*(pt) induced by the map to a point. This file proves the degree-zero part of the
statement that it detects the first nontrivial higher homotopy group π_k(X, x), k ≥ 2, of a
pointed space: the path components of Ω²X are the elements of π₂(X, x), so the reduced
zeroth homology H~₀(Ω²X; R) is the coproduct of copies of R indexed by the nontrivial
elements of π₂(X, x), the generator at a being [γ] - [1] for a double loop γ of class a.
If π₂(X, x) is nontrivial, H~₀(Ω²X; R) therefore contains a copy of R; for R = ℤ this is
an infinite cyclic subgroup.
For context, the positive-degree part is proved in other files. In positive degrees reduced and
ordinary homology agree (TauCeti.reducedSingularHomologySuccIso), and if π_j(X, x) vanishes
for 2 ≤ j < k and k ≥ 3, then H_{k-2}(Ω²X; ℤ) ≅ π_k(X, x) by the Hurewicz theorem for the
(k - 3)-connected space Ω²X: this is
TauCeti.MooreLoopSpace.doubleFirstHomologyAddEquiv for k = 3 and
TauCeti.MooreLoopSpace.doubleHomologyAddEquiv (or TauCeti.doubleLoopSpaceHomologyAddEquiv)
for k ≥ 4.
These are the homological facts behind the first part of Proposition 5.11 of
Barraud--Damian--Humilière--Oancea. In the paper's argument, the map
i_{Q*} : H_*(Q; i_Q^*𝓕) → H_*(𝓛₀Q; 𝓕), for the DG local system 𝓕 = C_*(Ω𝓛₀Q), is
identified with the map H_*(Ω²Q) → H_*(pt), whose kernel is H~_*(Ω²Q); that identification is
not formalized here, so the results of this file are statements about H~₀(Ω²Q) rather than about
ker i_{Q*}. The case of the sphere S² is
TauCeti.MooreLoopSpace.doubleReducedHomology₀SphereIso, with the homotopy groups of spheres.
Both the fixed-interval double loop space Ω (Ω X x) (Path.refl x) and the double Moore loop
space MooreLoopSpace (MooreLoopSpace X x) 1 are treated. The statements with coefficients in a
ring R take R and X in the same universe, so that R-modules live in the universe of the
space.
The path components of Ω²X and ΩᴹΩᴹX are identified with the elements of π₂(X, x) by
TauCeti.zerothHomotopyDoubleLoopSpaceEquiv and TauCeti.MooreLoopSpace.zerothHomotopyDoubleEquiv.
Main definitions #
TauCeti.doubleLoopSpaceReducedHomology₀Iso,TauCeti.MooreLoopSpace.doubleReducedHomology₀Iso:H~₀(Ω²X; R) ≅ ∐_{a ∈ π₂(X, x), a ≠ 1} R.
Main results #
TauCeti.exists_injective_doubleLoopSpaceReducedHomology₀,TauCeti.MooreLoopSpace.exists_injective_doubleReducedHomology₀: ifπ₂(X, x)is nontrivial, the reduced zeroth homology of the double loop space with coefficients in a ringRcontains a copy ofR, so an infinite cyclic subgroup forR = ℤ.
References #
- J.-F. Barraud, M. Damian, V. Humilière, A. Oancea, Floer homology with DG coefficients. Applications to cotangent bundles, arXiv:2404.07953, Proposition 5.11.
- A. Hatcher, Algebraic Topology, Proposition 2.7.
Reduced zeroth homology #
Reduced zeroth homology of the double loop space. H~₀(Ω²X; R) is the coproduct of
copies of R indexed by the nontrivial elements of π₂(X, x).
Instances For
The generator of TauCeti.doubleLoopSpaceReducedHomology₀Iso at the class of a double loop
γ is [γ] - [1].
The generator of TauCeti.doubleLoopSpaceReducedHomology₀Iso at the class of a double loop
γ is [γ] - [1].
Reduced zeroth homology of the double Moore loop space. H~₀(ΩᴹΩᴹX; R) is the coproduct of
copies of R indexed by the nontrivial elements of π₂(X, x).
Instances For
The generator of TauCeti.MooreLoopSpace.doubleReducedHomology₀Iso at the class of a double
Moore loop γ is [γ] - [1].
The generator of TauCeti.MooreLoopSpace.doubleReducedHomology₀Iso at the class of a double
Moore loop γ is [γ] - [1].
Statements with coefficients in a ring #
The degree-zero homological input to Proposition 5.11 of Barraud--Damian--Humilière--Oancea
(case k = 2). If π₂(X, x) is nontrivial, the reduced zeroth homology of the double loop
space with coefficients in R, the kernel of H₀(Ω²X; R) → H₀(pt; R), contains a copy of R; for
R = ℤ this is an infinite cyclic subgroup. The identification of this kernel with ker i_{Q*} is
not formalized here.
The degree-zero homological input to Proposition 5.11 of Barraud--Damian--Humilière--Oancea
(case k = 2), for Moore double loops. If π₂(X, x) is nontrivial, the reduced zeroth homology
of the double Moore loop space with coefficients in R contains a copy of R; for R = ℤ this is
an infinite cyclic subgroup.