Mapping cones and Homology groups

[math8]

Let $$W$$ be the mapping cone of the map $$f: S^{1} \rightarrow S^{1}$$ defined by $$f(z)= z^{p}$$

How do you compute the homology groups of $$W$$? What about the homology groups of the Universal covering of $$W$$?

I know that the mapping cone $$C_f$$ of $$f:X\rightarrow Y$$, is defined to be the quotient of the mapping cylinder of f and $$X$$. Or we can say,
Given a map $$f:X\rightarrow Y$$, the mapping cone $$C_f$$ is defined to be the quotient topological space of $$(X \times I) \sqcup Y$$ with respect to the equivalence relation $$(x, 0) \sim (x',0)\,, (x,1) \sim f(x)\,$$, on $$X$$. Here $$I$$ denotes the unit interval $$[0,1]$$ with its standard topology.

But I am not sure how to start using this definition of the mapping cone, to find the homology groups of $$W$$ and of its universal cover.

 

[mighty2000]

Can you please guide me through the steps?

To compute the homology groups of W, we can use the Mayer-Vietoris sequence. Let A = S^1 \times [0,1) and B = S^1 \times (0,1], then A \cup B = (S^1 \times [0,1)) \cup (S^1 \times (0,1]) = S^1 \times [0,1]. We can see that A and B are open sets covering W and their intersection is homotopy equivalent to the space S^1 \vee S^1 (the wedge sum of two circles).

Now, we can apply the Mayer-Vietoris sequence to the pair (W, S^1 \vee S^1) to get the long exact sequence:

...\rightarrow H_1(S^1 \vee S^1) \rightarrow H_0(W) \rightarrow H_0(S^1) \rightarrow H_0(S^1 \vee S^1) \rightarrow H_0(W) \rightarrow 0

Since H_1(S^1 \vee S^1) = 0 and H_0(S^1) = 0, we get:

0 \rightarrow H_0(W) \rightarrow H_0(S^1 \vee S^1) \rightarrow H_0(W) \rightarrow 0

This implies that H_0(W) \cong \mathbb{Z} \oplus \mathbb{Z}. Similarly, we can compute the other homology groups using the Mayer-Vietoris sequence and get:

H_1(W) \cong \mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}
H_i(W) = 0 for i \geq 2

To compute the homology groups of the universal cover of W, we can use the fact that the universal cover of W is homotopy equivalent to the infinite cylinder, which is contractible. Therefore, all of its homology groups are trivial, i.e. H_i(\widetilde{W}) = 0 for all i \geq 0.

To summarize, the homology groups of W are:

H_0(W) \cong \mathbb{Z} \oplus \mathbb{Z}
H_1(W) \con