Homology of spaces with weakly contractible path components

Let X be a space all of whose positive-dimensional homotopy groups vanish, at every base point: each path component of X is weakly contractible. Choosing a point in each path component gives a map to X from the set of path components with the discrete topology, which is a weak homotopy equivalence (ContinuousMap.isWeakHomotopyEquivalence_of_totallyDisconnectedSpace). The homology of a discrete space vanishes in positive degrees, so the same holds for X, with coefficients in any ring, for normalized cubical (TauCeti.isZero_homology_normalizedCubicalChainComplex_of_forall_subsingleton_homotopyGroup) and for singular homology (TauCeti.isZero_singularHomologyFunctor_of_forall_subsingleton_homotopyGroup). On chains: every cubical cycle of positive dimension of X is a boundary (TauCeti.NormalizedCubicalChain.exists_boundary_eq_of_forall_subsingleton_homotopyGroup).

The basic example is the loop space of an aspherical space, whose path components are the elements of the fundamental group.

Main results

• TauCeti.isZero_homology_normalizedCubicalChainComplex_of_forall_subsingleton_homotopyGroup, TauCeti.isZero_singularHomologyFunctor_of_forall_subsingleton_homotopyGroup: the positive-degree homology of a space with weakly contractible path components vanishes.
• TauCeti.NormalizedCubicalChain.exists_boundary_eq_of_forall_subsingleton_homotopyGroup: its cubical cycles of positive dimension are boundaries.

References

• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002, Proposition 4.21.

theorem TauCeti.isZero_homology_normalizedCubicalChainComplex_of_forall_subsingleton_homotopyGroup {X : Type u} [TopologicalSpace X] (R : Type u) [Ring R] (hX : ∀ (n : ℕ) (x : X), Subsingleton (HomotopyGroup.Pi (n + 1) X x)) {m : ℕ} (hm : m ≠ 0) : CategoryTheory.Limits.IsZero (HomologicalComplex.homology ((normalizedCubicalChainComplex R).obj ↧X) m)

A space with weakly contractible path components has no normalized cubical homology in positive degrees. If every positive-dimensional homotopy group of X vanishes, at every base point, then Hₘ(X; R) = 0 for m ≠ 0. The choice of a point in each path component is a weak homotopy equivalence from the discrete space of path components.

theorem TauCeti.isZero_singularHomologyFunctor_of_forall_subsingleton_homotopyGroup {X : Type u} [TopologicalSpace X] (R : Type u) [Ring R] (hX : ∀ (n : ℕ) (x : X), Subsingleton (HomotopyGroup.Pi (n + 1) X x)) {m : ℕ} (hm : m ≠ 0) : CategoryTheory.Limits.IsZero (((AlgebraicTopology.singularHomologyFunctor (ModuleCat R) m).obj ↧R).obj ↧X)

A space with weakly contractible path components has no singular homology in positive degrees. If every positive-dimensional homotopy group of X vanishes, at every base point, then Hₘ(X; R) = 0 for m ≠ 0.

theorem TauCeti.NormalizedCubicalChain.exists_boundary_eq_of_forall_subsingleton_homotopyGroup {X : Type u} [TopologicalSpace X] {R : Type u} [Ring R] (hX : ∀ (n : ℕ) (x : X), Subsingleton (HomotopyGroup.Pi (n + 1) X x)) {n : ℕ} {c : NormalizedCubicalChain (↑↧X) R (n + 1)} (hc : (boundary (↑↧X) R n) c = 0) : ∃ (c' : NormalizedCubicalChain (↑↧X) R (n + 2)), (boundary (↑↧X) R (n + 1)) c' = c

Cycles of a space with weakly contractible path components are boundaries. If every positive-dimensional homotopy group of X vanishes, at every base point, then every normalized cubical cycle of X of positive dimension is a boundary.