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Let $U$ and $V$ be affine algebraic varieties in $\mathbb{R}^k$ or $\mathbb{C}^k$. When is the Mayer-Vietoris sequence exact for $U \cup V$ and $U \cap V$? Intuitively, it feels as though the intersection may be "nice" enough so that $U$ and $V$ may be thickened into open sets in a way that will not change the homology of $U \cup V$ and $U \cap V$. Are there any references that one could suggest?

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    $\begingroup$ Just to make sure, you're asking about the Mayer-Vietoris sequence for singular homology, viewing everything in sight with the Euclidean topology, right? $\endgroup$
    – KReiser
    Oct 7, 2021 at 2:44
  • $\begingroup$ Yes, that is correct $\endgroup$
    – Powerspawn
    Oct 7, 2021 at 3:04

1 Answer 1

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Tools from semi-algebraic and subanalytic geometry can help us solve this problem. We may obtain a locally finite triangulation of $U\cup V$ which has as sub-triangulations $U$, $V$, and $U\cap V$. Combining this with Mayer-Vietoris for simplicial complexes and simplicial homology plus the equivalence between simplicial homology and singular homology of the geometric realization, we obtain the required result.

Theorem (Hironaka, Triangulations of Algebraic Sets in Algebraic Geometry, Arcata 1974 p.180): Given a locally finite system of subanalytic sets $\{X_\alpha\}$ in $\Bbb R^n$, there exists a simplicial decomposition $\Bbb R^n=\bigcup_a \Delta_a$ and a subanalytic automorphism $\kappa$ of $\Bbb R^n$ such that

  1. each $X_\alpha$ is a locally finite union of some of the $\kappa(\Delta_a)$ and
  2. $\kappa(\Delta_a)$ is a locally closed smooth real-analytic submanifold of $\Bbb R^n$ and $\kappa$ induces a real-analytic isomorphism $\Delta_a\stackrel{\sim}{\to}\kappa(\Delta_a)$ for every $a$.

This accomplishes our first goal: viewing complex varieties in $\Bbb C^m$ as real varieties in $\Bbb R^{2m}$ if necessary, we may take $\{U\cup V,U,V,U\cap V\}$ as our family to obtain a simultaneous triangulation of $U\cup V$, $U$, $V$, and $U\cap V$.

Next, we prove Mayer-Vietoris for simplicial complexes and simplicial homology. I'm sure this is written down somewhere you can reference, but I don't remember off the top of my head and a quick glance through the reference books I have at my desk didn't find it. (Plus the proof is short!)

Theorem. Let $K,L,M,N$ be simplicial complexes with $K=M\cup N$ and $L=M\cap N$. Then Mayer-Vietoris holds in simplicial homology, that is, there's a long exact sequence $$\cdots\to H_n^\Delta(L)\to H_n^\Delta(M)\oplus H_n^\Delta(N)\to H_n^\Delta(K) \to \cdots$$

Proof. Consider the short exact sequence of complexes $$0\to C_n(L)\to C_n(M)\oplus C_n(N)\to C_n(K)\to 0$$ where the first map is the map $x\mapsto (x,x)$ and the second map is $(x,y)\mapsto (x-y)$. Apply the snake lemma to obtain the long exact sequence. $\blacksquare$

Finally, we use the fact that simplicial homology of a simplicial set is the same as the singular homology of the geometric realization: this is theorem 2.27 in Allen Hatcher's Algebraic Topology, and is also covered here on MSE.

The combination of these results shows that Mayer-Vietoris always holds for singular homology of the analytic topology on affine algebraic varieties.

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  • $\begingroup$ So if I understand correctly, the essence of the argument is that $U \cup V$, $U$, $V$, and $U \cap V$ may be simultaneously triangulated, and the Mayer-Vietoris sequence always holds for simplicial complexes, correct? The inclusion-exclusion principle for the Euler characteristic also always holds for simplicial complexes, correct? $\endgroup$
    – Powerspawn
    Oct 25, 2021 at 7:02
  • $\begingroup$ Yes. ${}{}{}{}{}{}$ $\endgroup$
    – KReiser
    Oct 25, 2021 at 7:03

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