Homology for suspension

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Statement

In this article, we give the key results relating the homology groups of a topological space and the homology groups of its Suspension (?).

Version for unreduced homology

This states that:

H_{k+1}(SX) \cong H_k(X), k \ge 1

where H_k and H_{k+1} denote the k^{th} and (k+1)^{th} homology groups. The result also holds for homology groups with coefficients.

Further:

H_0(X) \cong H_1(SX) \oplus \mathbb{Z}

and:

H_0(SX) \cong \mathbb{Z}

Version for reduced homology

This states that:

\tilde{H}_{k+1}(SX) \cong \tilde{H}_k(X), k \ge -1

where \tilde{H}_k denotes the reduced homology. Note that for k \ge 1, reduced homology and unreduced homology coincide; for k = 0, the unreduced homology has an extra \mathbb{Z} in it. For k = -1, the right side is the trivial group, giving that \tilde{H}_0(SX) is trivial, so SX is a path-connected space.

Category-theoretic version

The isomorphisms between the homology groups of a topological space and its suspension are natural isomorphisms between these functors. In particular, if f:X \to Y is a continuous map, then we have an induced continuous map Sf:SX \to SY. There is a commuting diagram relating the homomorphism on k^{th} reduced homology between X and Y and the homomorphism on (k+1)^{th} reduced homology between SX and SY.

Facts used

  1. Mayer-Vietoris homology sequence

Proof

Recall that SX is obtained by taking X \times [0,1] (where [0,1] is the closed unit interval) and then identifying all points in X \times \{1 \} with each other and separately identifying all points in X \times \{ 0 \} with each other. We will call these two points p_1 and p_0 respectively. We consider the following open subsets U and V to use for the Mayer-Vietoris homology sequence:

Subset Concrete description Has a strong deformation retraction to ... More explanation
U SX \setminus \{ p_1 \} a point (i.e., it is contractible) SX \setminus \{ p_1 \} is homeomorphic to the cone space CX
V SX \setminus \{ p_0 \} a point (i.e., it is contractible) SX \setminus \{ p_0 \} is homeomorphic to the cone space CX
U \cap V X \times (0,1) X the factor (0,1) is a contractible space

Proof version with reduced homology

We note that U and V are open subsets and their union is X. Further, because of the strong deformation retraction facts mentioned, all reduced homology groups of U and V are trivial groups and all reduced homology groups of U \cap V are isomorphic to the corresponding reduced homology groups of X.

The original Mayer-Vietoris homology sequence reads:

\dots \to \tilde{H}_{k+1}(U) \oplus \tilde{H}_{k+1}(V) \to \tilde{H}_{k + 1}(SX) \to \tilde{H}_k(U \cap V) \to \tilde{H}_k(U) \oplus \tilde{H}_k(V) \to \dots

Every third term of this sequence is zero, aand the fragment above simplifies to:

\dots \to 0 \to \tilde{H}_{k+1}(SX) \to \tilde{H}_k(X) \to 0 \to \dots

Since this is a long exact sequence, the map \tilde{H}_{k+1}(SX) \to \tilde{H}_k(X) is forced to be an isomorphism. This completes the proof.