# What are the two different $\mathbb{S}^n$ in the construction of the homotopy group $\pi_n(\mathbb{S}^n)$ that classifies topological defects?

According to Mukhanov's Physical Foundations of Cosmology,

Homotopy groups give us a useful unifying description of topological defects. Maps of the $$n$$-dimensional sphere $$\mathbb{S}^n$$ into a vacuum manifold $$\mathcal{M}$$ are classified by the homotopy group $$\pi_n(\mathcal{M})$$. This group counts the number of topologically inequivalent maps from $$\mathbb{S}^n$$ into $$\mathcal{M}$$ that cannot be continuously deformed into each other.

For example, cosmic strings correspond to the homotopy group $$\pi_1(\mathcal{M})=\pi_1(\mathbb{S}^1)=\mathbb{Z}$$, which describes the maps of a one-dimensional sphere $$\mathbb{S}^1$$ into itself in a $${\rm U}(1)$$ theory.

Question In this map, one of the two $$\mathbb{S}^1$$ spaces (between which we consider the inequivalent maps) is the vacuum manifold of $${\rm U}(1)$$ given by $$|\phi|^2=\phi_1^2+\phi^2_2=v^2\tag{1}$$ where $$\phi=\phi_1+i\phi_2$$ and $$v$$ is a constant corresponding to the vacuum expectation value.

What is the other $$\mathbb{S}^1$$ in this case?

The source $$S^1$$ is, as is usual in the homotopical classification of defects, coming from considering a loop around a defect line (in this case a cosmic string).
• Okay. I guess this is a loop because the defect (cosmic string) is a one-dimensional curve. If the defect were pointlike or zero-dimensional, like the monopole, the appropriate shape to consider around it will be a sphere $\mathbb{S}^2$. Right? @NDewolf – SRS Dec 6 '20 at 10:08