Author: Eiko

Time: 2025-03-07 15:02:36 - 2025-03-07 15:02:36 (UTC)

Introduction To Sheaves

• A presheaf on $X$ with value in $\mathcal{C}$ is a contravariant functor $\mathcal{F}:X^{op}\to \mathcal{C}$, where $X$ is the category of open subsets in $X$ ordered by inclusion.

• Example, for $E\to X$ a vector bundle, the sections over $U$, $\mathcal{E}(U)$ form a sheaf of modules over $C^{\mathrm{\infty }}(U)$. Actually there is a categorical equivalence

  $$\text{Vector Bundles }(M)\stackrel{\text{take sections}}{\to }\text{Sheaves of }{C}^{\mathrm{\infty }}\text{-modules }(M)$$

• One example of a presheaf that is not a sheaf:

  The topological analogue of de-Rham complex, $C_X^k(U)=\text{Singular }k\text{-cochains on }U$ is a presheaf but not a sheaf. Why? Because if a cochain is zero on chains inside $U_1,U_2$, it does not uniquely glue: it can be non-zero on some very large chains that goes over the three regions $U_1-U_2,U_1\cap U_2,U_2-U_1$.

  The solution is to do sheafification, any presheaf can be sheafified.

Definition Of Manifold Using Sheaves

Let $(M,C_M^{\mathrm{\infty }})$ be a locally ringed space (actually $\mathbb{R}$-algebra) such that $\mathrm{\forall }x\in M$, there is $U\ni x$ such that $V\cong V\subset {\mathbb{R}}^n$ and it takes an isomorphism of functions

$$C_U^{\mathrm{\infty }}\cong i^{\ast }{C}_V^{\mathrm{\infty }}\in \mathbf{S}\mathbf{h}(U)$$

where $i:U\to V$ is the inclusion, and for $M$ to be a manifold, it is required to be paracompact, Hausdorff, and second countable.

Poincare Duality

Verdier duality is a cohomological duality in algebraic topology that generalizes Poincare duality for manifolds

Recall that Poincare duality claims that if $M$ is an $n$-dimensional oriented closed manifold, then the $k$-th cohomology group is isomorphic to the $n-k$-th homology group, i.e. for any coefficient ring that the orientation respects to,

$$H^k(M)\cong H_{n-k}(M):\alpha \mapsto [M]\cap \alpha$$

where $[M]$ is the fundamental class of $M$ and $\cap$ is the cap product.

Since orientation is trivial when taking mod $2$, you can drop the orientation condition for $R=\mathbb{Z}/2\mathbb{Z}$.

• In the case the manifold is not compact, we have to either

  • replace homology by Borel-Moore homology

    $$H^k(M)\cong H_{n-k}^{BM}(M)$$

  • or replace cohomology by compactly supported cohomology

    $$H_c^k(M)\cong H_{n-k}(M)$$

Ideas: Proofs of

• Poincare duality

  $$H^{\ast }(M)\stackrel{\sim }{\to }{H}_c^{n-k}(M{)}^{\ast }$$

• Kunneth theorem

  $$H^k(M_1\times M_2)\cong \underset{i+j=k}{⨁}{H}^i(M_1)\otimes H^j(M_2)$$

• De-Rham theorem

  $$H_{dR}^{\ast }(M,\mathbb{R})\cong H_{sing}^{\ast }(M,\mathbb{R})$$

• Using sheaves, everything reduces to contractible open coverings, Mayer-Vietoris sequence is actually a special case of such a Cech cohomology style argument.

• $$H_{dR}^k(M)\cong H_{sheaf}^k(M,\underset{―}{\mathbb{R}})\cong H_{sing}^k(M,\mathbb{R})$$

  where we used the resolution by soft sheaves (sections on closed $Z\subset M$ always extends to $M$.

  $$\underset{―}{\mathbb{R}}\to {\mathrm{\Omega }}_M^{\bullet }$$

Verdier Duality

• Take a local version of Poincare duality, $H^{n-k}(M,\mathbb{R})\cong \mathrm{Hom}(H_c^k(M,\mathbb{R}),\mathbb{R})$

• Verdier duality generalizes to $X\stackrel{f}{\to }Y$ of topological map,

  $$R\mathrm{Hom}({\mathcal{F}}^{\bullet },f^{!}{\mathcal{G}}^{\bullet })\cong R\mathrm{Hom}(R{f}_{!}{\mathcal{F}}^{\bullet },{\mathcal{G}}^{\bullet })$$

  $f_{!}$ are sections with compact support