Tools of (co)homology

 Preliminary exam prep

Let $X,Y$ be topological spaces, $G$ a group, and $R$ a unital commutative ring.

Defining homology groups

Theorem: If $(X,A)$ is a good pair (there exists a neighborhood $U\subset X$ of $A$ such that $U$ deformation retracts onto $A$), then for $i:A\hookrightarrow X$ the inclusion and $q:X\twoheadrightarrow X/A$ the quotient maps, there exists a long exact sequence of reduced homology groups
\[
\cdots \to \widetilde H_n(A) \tov{i_*} \widetilde H_n(X) \tov{q_*} \widetilde H_n(X/A) \to \cdots.
\]

Theorem: For any pair $(X,A)$, there exists a long exact sequence of homology groups
\[
\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to \cdots,
\]
where the last is called a relative homology group. Hence $H_n(X,A)\cong \widetilde H_n(X/A)$ for a good pair $(X,A)$.

Theorem (Excision): For any triple of spaces $(Z,A,X)$ with $\text{cl}(Z)\subset \text{int}(A)$, there is an isomorphism $H_n(X-Z,A-Z)\cong H_n(X,A)$.

For any $x\in X$, the local homology of $X$ at $x$ is the relative homology groups $H_n(X,X-\{x\})$. By excision, these are isomorphic to $H_n(U,U-\{x\})$ for $U$ any neighborhood of $x$. If $X$ is nice enough around $x$ (that is, if $U\cong \R^k$), then these groups are isomorphic to $H_n(\R^k,\R^k-\{x\})\cong H_n(D^k,\dy D^k) = H_n(S^k)$.

Theorem (Mayer-Vietoris): For $X=A\cup B$, there is a long exact sequence of homology groups
\[
\cdots \to H_n(A\cap B) \to H_n(A)\oplus H_n(B) \to H_n(X) \to \cdots,
\]

and if $A\cap B$ is non-empty, there is an analogous sequence for reduced homology groups.

Extending with coefficients

Recall the $\Tor$ and $\Ext$ groups, which were, respectively, the left and right derived functors of, respectively, $\otimes$ and $\Hom$ (see post "Exactness and derived functors," 2016-03-20). Here we only need $\Tor_1$ and $\Ext^1$, which are given by, for any groups (that is, $\Z$-modules) $A$, $B$,
\[
\begin{array}{r c c c l}
\Tor(A,B) & = & H_1(\text{projres}(A)\otimes B) & = & H_1(A\otimes \text{projres}(B)), \\
\Ext(A,B) & = & H^1(\Hom(A,\text{injres}(B))) & = & H^1(\Hom(\text{projres}(A),B)).
\end{array}
\]
Note that $\Tor$ is symmetric in its arguments, while $\Ext$ is not. Recall that $\Tor_0(A,B)=A\otimes B$ and $\Ext^0(A,B) = \Hom(A,B)$.

Theorem (Universal coefficient theorem): There exist isomorphisms
\[
\begin{array}{r c c c l}
H_n(X;G) & \cong & \Hom(H^n(X),G)\oplus \Ext(H^{n+1}(X),G) & \cong & H_n(X)\otimes G\ \oplus\ \Tor(H_{n-1}(X),G),  \\
H^n(X;G) & \cong & \Hom(H_n(X),G)\oplus \Ext(H_{n-1}(X),G) & \cong & H^n(X)\otimes G\ \oplus\ \Tor(H^{n+1}(X),G).
\end{array}
\]

Here are some common $\Hom$, $\Tor$, and $\Ext$ groups:
\begin{align*}
\Hom(\Z,G) & = G & \Tor(\Z,G) & = 0 & \Ext(\Z,G) & = 0 \\
\Hom(\Z_m,\Z) & = 0 & \Tor(G,\Z) & = 0 & \Ext(\Z_m,\Z) & = \Z_m \\
\Hom(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Tor(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Ext(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} \\
\Hom(\Q,\Z_n) & = 0 & & & \Ext(\Q,\Z_n) & = 0 \\
\Hom(\Q,\Q) & = \Q & & & \Ext(G,\Q) & = 0
\end{align*}

Theorem (Künneth formula): For $X,Y$ CW-complexes, $F$ a field, and $H^k(Y;G)$ or $H^k(X;G)$ finitely generated for all $k$, there are isomorphisms, for all $k$,

\[
H_k(X\times Y;F) \cong \bigoplus_{i+j=k} H_i(X;F)\otimes_FH_j(Y;F),
\hspace{1cm}
H^k(X\times Y;G) \cong \bigoplus_{i+j=k} H^i(X;G)\otimes_GH^j(Y;G)
\]

Dualities

Theorem (Poincaré duality): For $X$ a closed $n$-manifold (compact, without boundary) that is $R$-orientable (consistent choice of $R$-generator for each local homology group), for $k=0,\dots,n$ there are isomorphisms

\[
H^k(X;R)\cong H_{n-k}(X;R).
\]

Note that a simply orientable manifold means $\Z$-orientable. A manifold that is not $\Z$-orientable is always $\Z_2$-orientable (in fact all manifolds are $\Z_2$-orientable).

Theorem (Alexander duality): For $X\subsetneq S^n$ a non-empty closed locally contractible subset, for $k=0,\dots,n-1$ there are isomorphisms
\[
\widetilde H^k(X) \cong \widetilde H_{n-k-1}(S^n-X).
\]

References: Hatcher (Algebraic topology, Chapters 2, 3), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 7)

Degree and orientation

 Preliminary exam prep

Topology

 Recall that a topological manifold is a Hausdroff space in which every point has a neighborhood homeomorphic to $\R^n$ for some $n$. An orientation on $M$ is a choice of basis of $\R^n$ in each neighborhood such that every path in $M$ keeps the same orientation in each neighborhood. Every manifold $M\owns x$ appears in a long exact sequence (via relative homology) with three terms
\[
H_n(M-\{x\}) \tov{f} H_n(M) \tov{g} H_n(M,M-\{x\}).
\]
The first term is 0, because removing a point from an $n$-dimensional space leaves only its $(n-1)$-skeleton, which is at most $(n-1)$-dimensional. For $U$ a neighborhood of $x$ in $M$, the last term (via excision) is
\[
H_n(M-U^c, M-\{x\}-U^c) = H_n(U, U-\{x\})\cong H_n(\R^n, \R^n-\{x\}) \cong H_n(\R^n, S^{n-1}),
\]
which in turn fits into a long exact sequence whose interesting part is
\[
H_n(\R^n)\to H_n(\R^n,S^{n-1}) \to H_{n-1}(S^{n-1})\to H_{n-1}(\R^n),
\]
and since the first and last terms are zero, $H_n(M,M-\{x\})=\Z$. Since $f$ is zero, $g$ into $\Z$ must be injective, meaning that $H_n(M)=\Z$ or 0.

Theorem: Let $M$ be a connected compact (without boundary) $n$-manifold. Then

1. if $M$ is orientable, $g$ is an isomorphism for all $x\in M$, and
2. if $M$ is not orientable, $g=0$.

Definition: Let $f:M\to N$ be a map of connected, oriented $n$-manifolds. Since $H_n(M)=H_n(N)$ is infinite cyclic, the induced homomorphism $f_*:H_n(M)\to H_n(N)$ must be of the form $x\mapsto dx$. The number $d$ is called the degree of $f$.

In the special case when we are computing the degree for a map $f:S^n\to S^n$, by excision we get
\[
\deg(f) = \sum_{x_i\in f^{-1}(y)} \deg\left(H_n(U_i,U_i-x_i)\tov{f_*} H_n(V,V-y)\right),
\]for any $y\in Y$, some neighborhood $V$ of $y$, and preimages $U_i$ of $V$. This is called the local degree of $f$.

Geometry

Let $M$ be a smooth $n$-manifold. Recall $\Omega_M^r$ is the space of $r$-forms on $M$ and $d^r:\Omega^r_M\to \Omega^{r+1}_M$ is the differential map. Also recall the de Rham cohomology groups $H^r(M) = \text{ker}(d^r)/\text{im}(d^{r-1})$.

Definition: An $n$-manifold $M$ is orientable if it has a nowhere-zero $n$-form $\omega\in \Omega^n_M$. A choice of $\omega$ is called an orientation of $M$.

We also have a map $H^n(M)\to \R$, given by $\alpha\mapsto \int_M\alpha$, where the integral is normalized by the volume of $M$, so that integrating 1 across $M$ gives back 1. It is immediate that this doesn't make sense when $M$ is not compact, but when $M$ is compact and orientable, we get that $H^n(M)\neq0$. Indeed, if $\eta\in \Omega^{n-1}_M$ with $d\eta =\omega$, by Stokes' theorem we have
\[
\int_M\omega = \int_Md\eta = \int_{\dy M}\eta = \int_\emptyset\eta = 0,
\]
as $M$ has no boundary (since it is a manifold). But $\omega$ is nowhere-zero, meaning the first expression on the left cannot be zero. Hence $\omega$ is not exact and is a non-trivial element of $H^n(M)$.

Theorem: Let $M$ be a smooth, compact, orientable manifold of dimension $n$. Then $H^n(M)$ is one-dimensional.

Proof: The above discussion demonstrates that $\dim(H^n(M))\>1$. We can get an upper bound on the dimension by noting that the space of $n$-forms on $M$, given by $\Omega^n_M = \bigwedge^n(TM)^*$, has elements described by $dx_{i_1}\wedge \cdots \wedge dx_{i_n}$, with $\{i_1,\dots,i_n\}\subset \{1,\dots,n\}$. By rearranging the order of the $dx_{i_j}$, every element looks like $\alpha dx_1\wedge \cdots\wedge dx_n$ for some real number $\alpha$. Hence $\dim(\Omega^n_M) \leqslant 1$, so $\dim(H^n(M))$ is either 0 or 1. Therefore $\dim(H^n(M))=1$. $\square$

Definition: Let $f:M\to N$ be a map of smooth, compact, oriented manifolds of dimension $n$. Since $H^n(M)$ and $H^n(N)$ are 1-dimensional, the induced map $f^*:H^n(N)\to H^n(M)$ must be of the form $x\mapsto dx$. The number $d$ is called the degree of $f$. Equivalently, for any $\omega\in \Omega^n_N$,
\[
\int_M f^*\omega = d\int_N\omega
\]

References: Hatcher (Algebraic Topology, Chapters 2, 3.3), Lee (Introduction to Smooth Manifolds, Chapter 17)

The Eilenberg-Steenrod axioms

The category $\text{Top}$ of topological spaces may be generalized to the category $\text{Top}_*$ of pointed topological spaces. This in turn may be generalized to the category $\text{Top}_{rel}$ of pairs $(X,A)$, where $X\in\Obj(\text{Top})$ and $A$ is a subspace of $X$. The morphisms of $\text{Top}_{rel}$ on $(X,A)$ are the morphisms of $\text{Top}$ on $X$ paired with their restrictions to $A$. We write $(X)$ for $(X,\emptyset)$.

Definition 1: Let $X,Y\in\Obj(\text{Top}_*)$. Then $f\in\Hom_{\text{Top}_*}(X,Y)$ is an $n$-equivalence if the induced map on homotopy groups $f_*:\pi_k(X,x)\to \pi_k(Y,f(x))$ is an isomorphism for $k<n$ and an epimorphism for $k=n$. Further, $f$ is a weak equivalence if it is an $n$-equivalence for all $n\geqslant 1$. Similarly, $f\in \Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence if $f\in \Hom_{\text{Top}_*}(X,Y)$ and $f|_A\in \Hom_{\text{Top}_*}(A,B)$ are weak equivalences.

Definition 2: Let $C,D$ be two categories. A functor $\mathcal F:C\to D$ is an assignment $\mathcal F(X)\in \Obj(D)$ for every $X\in \Obj(C)$, and $\mathcal F(f)\in \Hom_D(\mathcal F(X),\mathcal F(Y))$ for every $f\in\Hom_C(X,Y)$. This assignment satisfies the following relations:
          $\mathcal F(g\circ f) = \mathcal F(g)\circ \mathcal F(f)$ for every $f\in \Hom_C(X,Y)$ and $g\in \Hom_C(Y,Z)$
          $\mathcal F(\id_X) = \id_{\mathcal F(X)}$ for every $X\in\Obj(C)$

Definition 3: Let $C$ be any category and $\mathcal F:\text{Top}\to C$ a functor. Then $\mathcal F$ is homotopy invariant if $f\simeq g$ in $\text{Top}$ implies $\mathcal F(f)=\mathcal F(g)$ in $C$, where $\simeq$ is the homotopy of maps.

Definition 4: A (relative) homology theory of topological spaces is a collection of homotopy-invariant functors $H_n:\text{Top}_{rel}\to \text{Ab}$ and a collection of natural transformations $d_n:H_n(X,A) \to H_{n-1}(A)$.

The Eilenberg-Steenrod axioms are properties a relative homology theory may satisfy. The number of axioms depends on how general a view of homology theories one would like. Eilenberg and Steenrod (7), May (4), Aguilar, Gitler, and Prieto (4), Wikipedia (5), and other sources (6,8) have all different numbers of axioms. The order of the axioms below is alphabetical.

For any $(X,A)\in\Obj(\text{Top}_{rel})$ and all $n$:

Axiom 1: Additivity. If $(X,A)=\bigoplus_i(X_i,A_i)$, then $H_n(X,A) \cong \bigoplus_iH_n(X_i,A_i),$ where the isomorphism is induced by the inclusions $(X_i,A_i)\hookrightarrow (X,A)$.

Axiom 2: Exactness. There is a long exact sequence
\[ \cdots \to H_{n+1}(X,A)\tov{d_{n+1}}H_n(A)\tov{\ \ }H_n(X)\tov{\ \ }H_n(X,A)\tov{d_n}H_{n-1}(A)\tov{\ \ }\cdots \]
where $H_n(A)\to H_n(X)$ and $H_n(X)\to H_n(X,A)$ are induced by the inclusions $(A)\hookrightarrow (X)$ and $(X)\hookrightarrow (X,A)$, respectively.

Axiom 3: Excision. If there exists a subset $U$ of $X$ with $\text{cl}(U)\subset \text{int}(A)$, then there is an isomorphism $H_n(X\setminus U,A\setminus U)\cong H_n(X,A)$ induced by the inclusion $(X\setminus U,A\setminus U)\hookrightarrow (X,A)$.

Axiom 4: Dimension. $H_n(*)=0$ for all $n\neq 0$.

Axiom 5: Weak equivalence. If $f\in\Hom_{\text{Top}_{rel}}((X,A),(Y,B))$ is a weak equivalence, then the induced map on homology $f_*:H_n(X,A)\to H_n(Y,B)$ is an isomorphism.

Singular homology is a homology theory that satisfies all the axioms above. $K$-theory is a homology theory that does not satisfy the dimension axiom.

References: May (A Concise course in Algebraic Topology, Chapter 13.1), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 5.3)