代数拓扑Higher Homotopy Groups and A Connection to Cohomology

jun

Abstract:
The definition for homotopy groups is easy to understand, but the groups are very difficult to compute. On the other hand, cohomology groups have a rather abstract and indirect definition, but they are relatively easier to work with. However, there is a theorem pointing out that there exists a natural bijective correspondence between some homotopy classes and cohomology groups. This gives a second, completely different perspective on cohomology.


Homotopy groups:

In algebraic topology, we try to attach algebraic invariants to spaces to distinguish a space from another. One example of the algebraic invariants are the homotopy groups. In the remaining paragraphs, all functions are continuous unless stated otherwise.

An [tex]n-[/tex]dimensional sphere can be represented by the quotient space [tex]D^n / {\partial D^n}[/tex]. In this way, for a space [tex]X[/tex], any map [tex]f: D^n \to X[/tex] with the restriction [tex]f(\partial D^n )= x_0 \in X[/tex] for some fixed point [tex]x_0 \in X[/tex] induces a map from the [tex]n-[/tex]dimensional sphere to the space [tex]X[/tex]. The homotopy classes of these maps, i.e. the set containing every [tex][f][/tex]'s, forms a group with the following law of composition:

$ (f+g)(s_1, s_2, ... , s_n)= $
$f(2s_1 , s_2, ... , s_n)$ when $s_1 \in [0, 1/2]$;      
$g(2 s_1 -1, s_2, ..., s_n)$ when $ s_1 \in [1/2 , 1]$.

The above group structure, for each natural number $n$, is the $n$th homotopy group of $X$ and is denoted by $\pi_n (X, x_0)$.

It is shown in Hatcher (p. 28) that if a space is connected, then choosing any point as the base point in $X$ does not affect the homotopy groups structure. This can be proven in the following way. Suppose the two base points $x_0 , x_1 \in X$ are connected by the path $l$. For a representing $f_0$ of an element of $\pi_n (X,x_0)$, we can continuously transform $f_0$ to $f_1$ such that $f_1$ sends an open neighbourhood of $\partial D^n$ to $l$ and $\partial D^n$ to $x_1$. It can be verified that $f_1$ is a representative of an element in $\pi_n (X,x_1)$. In this way, elements in $\pi_n (X,x_0)$ corresponds to those in $\pi_n (X,x_1)$. And this correspondence induces an isomorphism of the group structure for each $n$. Hence for a connected space, we can simply denote the $n$th homotopy group as $\pi_n (X)$. It can also be seen that a base point preserving map $\phi: (X, x_0) \to (Y, y_0)$ induces a group homomorphisms $\phi_\star : \pi_n (X, x_0) \to \pi_n(Y, y_0)$.

We can extend the definitions a little bit to define the relative homotopy groups $\pi_n (X, A, x_0)$ with $x_0\in A \subset X$. One way to define this is the homotopy classes of functions $f: D^n \to X$ such that $f(\partial D^n) \subset A$ and $f( {d_0}) = x_0$ for some $d_0 \in \partial D^n$. We use $i,j$ to denote the inclusion maps $i: (A, x_0) \rightarrow (X, x_0)$ and $j: (X, x_0, x_0) \rightarrow (X, A, x_0)$. Also, we use $\partial$ to denote the restriction of functions $f: (D^n, \partial D^n,{d_0})\to (X,A,x_0)$ to $\partial D^n$. The first long exact sequence we see about higher homotopy groups is given by the following theorem.

Theorem 1 (Hatcher p. 344) Let $i_\star$ and $j_\star$ denote the induced group homomorphism of $i, j$. There is a long exact sequence:
$... \to \pi_n (A, x_0) \stackrel{i_\star}{\rightarrow} \pi_n(X, x_0) \stackrel{j_\star}{\rightarrow} \pi_n (X,A,x_0) \stackrel{\partial}{\rightarrow} \pi_{n-1} (A, x_0) \to ... \to \pi_0(X, x_0) $.


CW complexes are one type of the "nicer'' spaces commonly consided in Algebraic Topology. They are constructed by inductively attaching higher dimensional cells to the previously structures. There are many nice properties that CW complexes have. For instance (Hatcher p. 349), every map between CW complexes is homotopic to a cellular map (i.e. alway mapping cells to the same or lower dimension). Whitehead's Theorem about CW complexes, is one of the results that shows why studying the higher homotopy groups are important.

Theorem 2 (Whitehead's Theorem, Hatcher p. 348) If a map $f: X \to Y$ between connected CW complexes induces isomorphism $f_\star : \pi_n(X) \to \pi_n(Y)$ for all n, then $f$ is a homotopy equivalence. In the case $f$ is the inclusion map of a subcomplex $X\rightarrow Y$, the conclusion is stronger: $X$ is a deformation retraction of $Y$.

Whitehead's Theorem can be proven using the following useful lemma called the Compression Lemma.

Lemma 3 (Compression Lemma Hatcher p. 348) Let $(X, A)$ be a CW pair and let $(Y, B)$ be any pair with $B \ne \emptyset$. For each $n$ such that $X-A$ has cells of dimension $n$, assume that $\pi_n(Y,B,y_0)=0$ for all $y_0 \in B$. Then every map $f: (X,A) \to (Y,B)$ is homotopic rel$A$ to a map $X\to B$.

Connection to Cohomology
Higher homotopy groups for arbitrary spaces turn out to be very difficult to compute. But there are some interesting results that show homotopy groups are actually related to cohomology groups in a subtle way.

The Eilenberg-MacLane Spaces, denoted by $K(G,n)$ is a family of spaces that has the following property:

for every abelian group $G$,
$\pi_i (K (G,n))=$     
$0$ if $i \ne n$;      
$G$ if $i = n$

These spaces can be constructed first by constructing a space $X$ with $\pi_i (X) =0$ $\forall i<n$ and $\pi_n(X) = G$. Then for each nontrivial element $[f] \in \pi_j (X)$ for some $j> n$, we can attach a cell $e^{j+1}$ by evaluating the boundary of the cell through $f$. It can be shown that this construction does not affect the previous homotopy groups while ``deleting'' the nontrivial element $[f]$. By repeating this procedure, $K(G,n)$ can be constructed.

The spaces $K(G,n)$ turns out to be closely related to the cohomology groups $H^n (X;G)$. There are many definitions for a (reduced) cohomology theory, and usually the definition is more complicated than the one for homotopy groups. The following theorem (Hatcher p. 393) shows that there is a natural bijections from the homotopy classes $[X, K(G,n)]$ to $H^n (X;G)$ if $X$ is a CW complex.

Theorem 4 (Hatcher p. 393 theorem 4.57) There are natural bijections $T: [X, K(G,n)]\to H^n (X;G)$ for all CW complexes $X$ and all $n >0$, with $G$ any abelian group. Such a $T$ has the form $T([f])=f_\star (\alpha)$ for a certain distinguished class $\alpha \in H^n(K(G,n);G)$.

The theorem can be proved by verifying that the assignment of groups satisfy the axioms for a (reduced) cohomology theory. In the proof, one of the critical step is to show the assignment $h^{n+1} (\Sigma X) = h^n (X)$. This follows from the fact that $[\Sigma X,Y] \cong [X, \Omega Y]$ and $\Omega K(G, n+1) = K(G, n)$.

This result gives a totally different perspective on cohomology. To understand cohomology groups with coefficients in $G$, it is enough just to study the space $K(G,n)$. And the result, on the other hand, shows that homotopy groups and cohomology groups for a space $X$ are very much related, with the the former one being homotopy classes of maps into $X$, and the later one being homotopy classes of maps from $X$ into other spaces.

Acknowledgement
I want to thank Vigleik Angeltveit for his help and supervising the project. This report is supported by Summer Research Scholarship at ANU.


Reference

A. Hatcher, (2002), Algebraic Topology, Cambridge University Press.