# Homotopy Type of Polynomial Hypersurfaces

It is known that the homotopy type of the locus $\{ (z_0, \dots, z_n) \in \mathbb{C} \, | \, z_0^{a_0} + \cdots + z_n^{a_n} = 1 \}$, where $a_0, \dots, a_n \in \mathbb{N}$, is a $\mu$-fold wedge sum of $n$-spheres, where $\mu = \prod_{i = 1}^{n} (a_i - 1)$. These spaces are the diffeomorphic to the fibers of Brieskorn-Pham Manifolds $\Sigma(a_0, \dots, a_n)$.

I'd like to understand related spaces defined by a single monomial in several variables. Consider the locus of points $V_n = \{ (z_0, \dots, z_n) \in \mathbb{C} \, | \, z_0 \cdots z_n = 1 \}$. Since $z_k = (z_1 \cdots \hat{z}_{k} \cdots z_n)^{-1}$ for $0 \leqslant k \leqslant n$, then $V_{n}$ is diffeomorphic to the product space $(\mathbb{C}^{\times})^{n}$, which has the homotopy type of a torus $\mathbb{T}^{n} = (S^{1})^{n}$, that is, $V_{n} \simeq \mathbb{T}^{n}$. Similarly, consider the locus of points $V_{n,m} = \{ (z_0, \dots, z_n) \in \mathbb{C} \, | \, (z_0 \cdots z_n)^{m} = 1 \}$ Since there are $m$ $m^{\text{th}}$-roots of unity, one can show that $V_{n,m}$ is diffeomorphic to the $m$-fold disjoint union of $V_{n} \cong (\mathbb{C}^{\times})^{n} \simeq \mathbb{T}^{n}$, that is, $V_{n,m} \simeq \bigsqcup_{i = 1}^{m} \mathbb{T}^{n}$.

I suspect the homotopy type of the locus $V_{n, (k_0, \dots, k_n)} = \{ (z_0, \dots, z_n) \in \mathbb{C} \, | \, z_0^{k_0} \cdots z_n^{k_n} = 1 \}$ for arbitrary $k_0, \dots, k_n \in \mathbb{N}$ is the $d = \gcd(k_1, \dots, k_n)$-fold disjoint union of that of the reduced locus $V_{n, (k_0/d, \dots, k_n/d)} = \{ (z_0, \dots, z_n) \in \mathbb{C} \, | \, z_0^{k_0/d} \cdots z_n^{k_n/d} = 1 \}$, so it suffices to consider those spaces where $\gcd(k_0, \dots, k_n) = 1$.

What is the homotopy type of the space $V_{n, (k_0, \dots, k_n)}$ where $\gcd(k_0, \dots, k_n) = 1$? Does this generalize to sums of monomials with common variables?

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Let $\sigma:\mathbb{C}^{n+1}/\mathbb{Z}^{n+1}\rightarrow (\mathbb{C}^\times)^{n+1}$, $\sigma(w_0,\cdots, w_n)=(e^{2\pi iw_0},\cdots,e^{2\pi iw_n})$ be the standard biholomorphic map, then $\sigma^{-1}(V_{n,(k_i)})=(k_0w_0+\cdots+k_nw_n=0)\subset \mathbb{C}^{n+1}/\mathbb{Z}^{n+1}$ which is a quotient of hyperplane in $\mathbb{C}^{n+1}$.

Now we want to analyze the behaviour of $\mathbb{Z}^{n+1}$ acting on the hypersurface $(k_0w_0+\cdots+k_nw_n=0)\subset \mathbb{C}^{n+1}$. For any $A\in SL(n+1,\mathbb{Z})$, $A$ induces a biholomorphic map of $\mathbb{C}^{n+1}/\mathbb{Z}^{n+1}$ naturally by matrix multiplication. Thus $\sigma\circ A\circ\sigma^{-1}$ defines a biholomorphic map of $(\mathbb{C}^\times)^{n+1}$. So we want to choose a suitable $A$ to transform $(k_0w_0+\cdots+k_nw_n=0)$ into a simpler form.

If $\gcd(k_0,\cdots,k_n)=1$, then we can choose $A\in SL(n+1,\mathbb{Z})$ such that the first row of $A$ is exactly $(k_0,\cdots,k_n)$, which means that if $A(w_0,\cdots,w_n)^t=(w_0',\cdots,w_n')^t$, then $w_0'=k_0w_0+\cdots+k_nw_n$. Therefore, $A\circ\sigma^{-1}(V_{n,(k_i)})=(w_0'=0)$, so $\sigma\circ A\circ\sigma^{-1}(V_{n,(k_i)})=(z_0'=1)\cong (\mathbb{C}^\times)^{n}\subset (\mathbb{C}^\times)^{n+1}$.

So we have the conclusion: $V_{n,(k_i)}\cong(\mathbb{C}^\times)^{n}$, which implys the homotopy type of it is again $\mathbb{T}^n$.

Remark: Here $V_{n,(k_i)}$ is a complex subgroup of $(\mathbb{C}^\times)^{n+1}$, so we can describe it simply by passing to $\mathbb{C}^{n+1}/\mathbb{Z}^{n+1}$.

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