用户: Eee/Hatcher

For a quick review of Homology Theory, one can refer to the Appendix of Characteristic Classes written by Milnor and Stasheff.

0Some Underlying Geometric Notions

Definition 0.1. A retraction of $X$ onto a subspace $A$ is a map $r:X\to A$ such that $r|A=\mathrm{id}$.
A deformation retraction of $X$ onto $A$ is a homotopy from ${\mathrm{id}}_X$ to $r$.

Definition 0.2. For a subspace $A\subset X_1$ and a map $f:A\to X_0$, $X_0{\bigsqcup }_f{X}_1$, ${\mathbf{X}}_0$ with ${\mathbf{X}}_1$ attached along $\mathbf{A}$ via $\mathbf{f}$, is the quotient space $X_0\bigsqcup X_1/\{a\in A\sim f(a)\in X_0\}$.

Definition 0.3 (Operations on Spaces).
The cone $CX=X\times I/X\times \{0\}$.
The suspension $SX=X\times I/X\times \{0\}/X\times \{1\}$.
The mapping cylinder $M_f$ is $X\times I\supset X\times \{1\}\stackrel{f}{\to }Y$.
The mapping cone $C_f$ is $CX\supset X\times \{1\}\stackrel{f}{\to }Y$.
The join $X\ast Y=X\times Y\times I/\{(x,y,0)\sim (x,y^{\prime },0),(x,y,1)\sim (x^{\prime },y,1)\}$, that is, collapsing $X\times Y\times \{0\}$ to $X$ and $X\times Y\times \{1\}$ to $Y$.
The wedge sum $X\vee Y=X\bigsqcup Y/\{x_0\sim y_0\}$ for chosen points $x_0\in X,y_0\in Y$.
The smash product $X\wedge Y=X\times Y/X\ast Y$.

Definition 0.4. A cell complex or a CW complex $X$ is a space obtained by $X^0\subset X^1\subset \cdots$, starting from a set of points $X_0$ and inductively forming the $n$-skeleton $X^n$ by attaching $n$-cells $e_{\alpha }^n\supset S^{n-1}\to X^{n-1}$. Either set $X=X^n$ stopping at some finite $n$, or set $X=\bigcup _n{X}^n$ giving the weak topology: $A\subset X$ is open iff $A\cap X^n$ is open for each $n$.

Definition 0.5. A subcomplex of $X$ is a closed subspace $A$ that is a union of cells of $A$. $(X,A)$ is called a CW pair.

Example 0.6. A graph is $X=X^1$, vertices with edges attached, two ends of which can be attached to the same vertex.

Example 0.7. The sphere $S^n=D^n/\partial D^n$ has the structure $e^n\supset S^{n-1}\to e^0$.

Example 0.8. Real projective $\mathbf{n}$-space $\mathbb{R}{P}^n=S^n/\{v\sim -v\}$ is also the quotient space of a semisphere $D^n$ with antipodal points identified (just $\mathbb{R}{P}^{n-1}$), or $e^n\supset S^{n-1}\to \mathbb{R}{P}^{n-1}$. So $\mathbb{R}{P}^n=e^0\cup e^1\cup \cdots \cup e^n$.

Example 0.9. Complex projective $\mathbf{n}$-space $\mathbb{C}{P}^n=e^0\cup e^2\cup \cdots \cup e^{2n}$.

Definition 0.10. Orientable surface $M_g$

 Attach $2$-cell to the wedge sum of $2g$ circles $a_1,b_1,\cdots a_g,b_g$ by the word $[a_1,b_1]\cdots [a_g,b_g]=a_1{b}_1{a}_1^{-1}{b}_1^{-1}\cdots a_g{b}_g{a}_g^{-1}{b}_g^{-1}$ we obtain an orientable surface $M_g$.

Definition 0.11. Nonorientable surface $N_g$

 Attach $2$-cell to the wedge sum of $g$ circles $a_1,\cdots a_g$ by the word $a_1^2\cdots a_g^2$ we obtain a nonorientable surface $N_g$.

 For example, $N_1$ is the projective plane $\mathbb{R}{P}^2$, and $N_2$ is the Klein bottle.

1Fundamental Group

 One can often show that two spaces are not homeomorphic by showing that their fundamental groups are not isomorphic, since it will be easy consequence of the definition of the fundamental group that homeomorphic spaces have isomorphic fundamental groups.

Basic Constructions

Definition 1.1. A homotopy of paths in $X$ is a family $f_t:I\to X,0\le t\le 1$, such that

  1. The endpoints $f_t(0)=x_0$ and $f_t(1)=x_1$ are independent of $t$.

  2. The associated map $F:I\times I\to X$ defined by $F(s,t)=f_t(s)$ is continuous.

 When two paths $f_0$ and $f_1$ are connected in this way by a homotopy $f_t$, they are said to be homotopic. The notation for this is $f_0\sim f_1$.

Definition 1.2. The relation of homotopy on paths with fixed endpoints in any space is an equivalence relation.

 The equivalence class of a path $f$ under the equivalence relation of homotopy will be denoted $[f]$ and called the homotopy class of $f$.

Definition 1.3. The set of all homotopy classes $[f]$ of loops $f:I\to X$ at the basepoint $x_0$ is denoted ${\pi }_1(X,x_0)$.

 ${\pi }_1(X,x_0)$ is a group with respect to the product $[f][g]=[f\cdot g]$. This group is called the fundamental group of $X$ at the basepoint $x_0$.

Definition 1.4. Define a reparametrization of a path $f$ to be a composition $f\phi$ where $\phi :I\to I$ is any continuous map such that $\phi (0)=0$ and $\phi (1)=1$.

 Reparametrizing a path preserves its homotopy class since $f\phi \sim f$ via the homotopy $f{\phi }_t$, where ${\phi }_t=(1-t)\phi (s)+ts$.

Definition 1.5. Say $h$ is a path from $x_0$ to $x_1$, then we define a change-of-basepoint map ${\beta }_h:{\pi }_1(X,x_1)\to {\pi }_1(X,x_0)$ by ${\beta }_h[f]=[h\cdot f\cdot \overline{h}]$.

Proposition 1.6. The map ${\beta }_h:{\pi }_1(X,x_1)\to {\pi }_1(X,x_0)$ is an isomorphism.

 Thus if $X$ is path-connected, the group ${\pi }_1(X,x_0)$ is, up to isomorphism, independent of the choice of basepoint $x_0$. In this case the notation ${\pi }_1(X,x_0)$ is often abbreviated to ${\pi }_1(X)$.

Definition 1.7. A space is called simply-connected if it is path-connected and has trivial fundamental group.

Proposition 1.8. A space $X$ is simply-connected iff there is a unique homotopy class of paths connecting any two points in $X$.

Definition 1.9. Given a space $X$, a covering space of $X$ consists of a space $\tilde{X}$ and a map $p:\tilde{X}\to X$ satisfying the following condition:

  For each point $x\in X$, there is an open neighborhood $U$ of $x$ in $X$ such that $p^{-1}(U)$ is a union of disjoint open sets each of which is mapped homeomorphically onto $U$ by $p$.

 Such a $U$ will be called evenly covered.

 There are three easy lifting properties.

Proposition 1.10. For each path $f:I\to X$ starting at a point $x_0\in X$ and each $\widetilde{x_0}\in p^{-1}(x_0)$ there is a unique lift $\tilde{f}:I\to \tilde{X}$ starting at $\widetilde{x_0}$.

Proposition 1.11. For each homotopy $f_t:I\to X$ of paths starting at $x_0$ and each $\widetilde{x_0}\in p^{-1}(x_0)$ there is a unique lifted homotopy $\widetilde{f_t}:I\to \tilde{X}$ of paths starting at $\widetilde{x_0}$.

Proposition 1.12. Given a map $F:Y\times I\to X$ and a map $\tilde{F}:Y\times \{0\}\to \tilde{X}$ lifting $F{|}_{Y\times \{0\}}$, then there is a unique map $\tilde{F}:Y\times I\to \tilde{X}$ lifting $F$ and restricting to the given $\tilde{F}$ on $Y\times \{0\}$.

Theorem 1.13. $\pi (S^1)$ is an infinite cyclic group generated by the homotopy class of the loop $\omega (s)=(\cos 2\pi s,\sin 2\pi s)$ based at $(1,0)$.

 In other words, $\pi (S^1)\cong \mathbb{Z}$.

Theorem 1.14. Algebraic fundamental theorem

 Every nonconstant polynomial with coefficients in $\mathbb{C}$ has a root in $\mathbb{C}$.

Theorem 1.15. Brouwer fixed point theorem in dimension $2$

 Every continunous map $h:D^2\to D^2$ has a fixed point, that is, a point $x\in D^2$ with $h(x)=x$.

Remark 1.16. Brouwer fixed point theorem in dimension $n$

 Every continunous map $h:D^n\to D^n$ has a fixed point, that is, a point $x\in D^n$ with $h(x)=x$.

Theorem 1.17. Borsuk-Ulam theorem in dimension $2$

 For every continuous map $f:S^2\to {\mathbb{R}}^2$ there exists a pair of antipodal points $x$ and $-x$ in $S^2$ with $f(x)=f(-x)$.

Remark 1.18. Borsuk-Ulam theorem in dimension $n$

 For every continuous map $f:S^n\to {\mathbb{R}}^n$ there exists a pair of antipodal points $x$ and $-x$ in $S^n$ with $f(x)=f(-x)$.

Corollary 1.19. There is no one-to-one continuous map from $S^2$ to ${\mathbb{R}}^2$, so $S^2$ is not homeomorphic to a subspace of ${\mathbb{R}}^2$.

Corollary 1.20. Whenever $S^2$ is expressed as the union of three closed sets $A_1,A_2$ and $A_3$, then at least one of these sets must contain a pair of antipodal points $\{x,-x\}$.

 Projecting the four faces of the tetrahedron radially onto the sphere, we obtain a cover of $S^2$ by four closed sets, none of which contains a pair of antipodal points.

Remark 1.21. Assuming the higher-dimensional version of the Borsuk-Ulam theorem, the same arguments show that $S^n$ cannot be covered by $n+1$ closed sets without antipodal pairs of points, though it can be covered by $n+2$ such sets, as the higher-dimensional analog of a tetrahedron shows.

Proposition 1.22. ${\pi }_1(X\times Y)\cong {\pi }_1(X)\times {\pi }_1(Y)$ if $X$ and $Y$ are path-connected.

Example 1.23. The Torus: ${\pi }_1(S^1\times S^1)\cong Z\times Z$.

Definition 1.24. Suppose $\phi :X\to Y$ is a map taking the basepoint $x_0\in X$ to the basepoint $y_0\in Y$. For brevity we write $\phi :(X,x_0)\to (Y,y_0)$ in this situation.

 Then $\phi$ induces a homomorphism ${\phi }_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$ defined by composing loops $f:I\to X$ based at $x_0$ with $\phi$, that is, ${\phi }_{\ast }[f]=[\phi f]$.

Proposition 1.25. Two basic properties of induced homomorphisms are:

  1. $(\phi \psi {)}_{\ast }=(\phi {)}_{\ast }(\psi {)}_{\ast }$ for a composition $(X,x_0)\stackrel{\psi }{\to }(Y,y_0)\stackrel{\phi }{\to }(Z,z_0)$.

  2. $1_{\ast }=1$, which is a concise way of saying that the identity map $1:X\to X$ induces the identity map $1:{\pi }_1(X,x_0)\to {\pi }_1(X,x_0)$.

 These two properties of induced homomorphisms are what makes the fundamental group a functor.

Lemma 1.26. The decomposition of loops subordinated to an open covering.

 If a space $X$ is the union of a collection of path-connected open sets $A_{\alpha }$ each containing the basepoint $x_0\in X$ and if each intersection $A_{\alpha }\cap A_{\beta }$ is path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_{\alpha }$.

Theorem 1.27. ${\pi }_1(S^n)=0$ if $n\ge 2$.

Corollary 1.28. ${\mathbb{R}}^2$ is not homeomorphic to ${\mathbb{R}}^n$ for $n\ne 2$.

Remark 1.29. The more general statement that ${\mathbb{R}}^m$ is not homeomorphic to ${\mathbb{R}}^n$ if $m\ne n$ can be proved in the same way using either the higher homotopy groups or homology groups.

Proposition 1.30. If a space $X$ retracts onto a subspace $A$, then the homomorphism $i_{\ast }:{\pi }_1(A,x_0)\to {\pi }_1(X,x_0)$ induced by the inclusion $i:A\hookrightarrow X$ is injective. If $A$ is a deformation retract of $X$, then $i_{\ast }$ is an isomorphism.

Definition 1.31. A basepoint-preserving homotopy ${\phi }_t:(X,x_0)\to (Y,y_0)$ is the case that ${\phi }_t(x_0)=y_0$ for all $t$.

 Similarly, we can define homotopy equivalence for spaces with basepoint. One says $(X,x_0)\sim (Y,y_0)$ if there are maps $\phi :(X,x_0)\to (Y,y_0)$ with homotopies $\phi \psi \sim 1$ and $\psi \phi \sim 1$ through maps fixing the basepoints.

Proposition 1.32. If ${\phi }_t:(X,x_0)\to (Y,y_0)$ is a basepoint-preserving homotopy, then ${\phi }_{0\ast }={\phi }_{1\ast }$.

Proposition 1.33. If $(X,x_0)\sim (Y,y_0)$, then ${\pi }_1(X,x_0)\cong {\pi }_1(Y,y_0)$.

 For homotopy equivalences one does not have to be quite so careful, as the conditions on basepoints can actually be dropped.

Proposition 1.34. If $\phi :X\to Y$ is a homotopy equivalence, then the induced homomorphism ${\phi }_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,\phi (x_0))$ is an isomorphism for all $x_0\in X$.

Lemma 1.35. If ${\phi }_t:X\to Y$ is a homotopy and $h$ is the path ${\phi }_t(x_0)$ formed by the images of a basepoint $x_0\in X$, then we have ${\phi }_{0\ast }={\beta }_h{\phi }_{1\ast }$.

Van Kampen’s Theorem

 The Van Kampen theorem gives a method for computing the fundamental groups of spaces that can be decomposed into simpler spaces whose fundamental groups are already known.

 We shall see for example that for every group $G$ there is a space $X_G$ whose fundamental group is isomorphic to $G$.

Definition 1.36. Free product. Refer to Hatcher P41.

Definition 1.37. Free group is the free product of any number of copies of $\mathbb{Z}$, finite or infinite. The generators are called a basis for the free group, and the number of basis element is the rank of the free group.

Definition 1.38. A basic property of the free product ${\ast }_{\alpha }{G}_{\alpha }$ is that any collection of homomorphisms ${\phi }_{\alpha }:G_{\alpha }\to H$ extends uniquely to a homomorphism $\phi :{\ast }_{\alpha }{G}_{\alpha }\to H$.

 Namely, the value of $\phi$ on a word $g_1\cdots g_n$ with $g_i\in G_{{\alpha }_i}$ must be ${\phi }_{{\alpha }_1}(g_1)\cdots {\phi }_{{\alpha }_n}(g_n)$, and using this formula to define $\phi$ gives a well-defined homomorphism.

 Suppose a space $X$ is decomposed as the union of a collection of path-connected open subsets $A_{\alpha }$, each of which contains the basepoint $x_0\in X$. The homomorphisms $j_{\alpha }:{\pi }_1(A_{\alpha })\to {\pi }_1(X)$ induced by the inclusions $A_{\alpha }\hookrightarrow X$ extend to a homomorphism $\Phi :{\ast }_{\alpha }{\pi }_1(A_{\alpha })\to {\pi }_1(X)$.

 The Van Kampen theorem will say that $\Phi$ is very often surjective, but we can expect $\Phi$ to have a nontrivial kernel in general.

 For $i_{\alpha \beta }:{\pi }_1(A_{\alpha }\cap A_{\beta })\to {\pi }_1(A_{\alpha })$ is the homomorphism induced by the inclusion $A_{\alpha }\cap A_{\beta }\hookrightarrow A_{\alpha }$ then $j_{\alpha }{i}_{\alpha \beta }=j_{\beta }{i}_{\beta \alpha }$, both these compositions being induced by the inclusion $A_{\alpha }\cap A_{\beta }\hookrightarrow X$, so the kernel of $\Phi$ contains all the elements of the form $i_{\alpha \beta }(\omega )i_{\beta \alpha }(\omega {)}^{-1}$ for $\omega \in {\pi }_1(A_{\alpha }\cap A_{\beta })$.

Theorem 1.39. Van Kampen theorem

 If $X$ is a union of path-connected open sets $A_{\alpha }$ each containing the basepoint $x_0\in X$ and if each intersection $A_{\alpha }\cap A_{\beta }$ is path-connected, then the homomorphism $\Phi :{\ast }_{\alpha }{\pi }_1(A_{\alpha })\to {\pi }_1(X)$ is surjective.

 If in addition each intersection $A_{\alpha }\cap A_{\beta }\cap A_{\gamma }$ is path-connected, then the kernel of $\Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{\alpha \beta }(\omega )i_{\beta \alpha }(\omega {)}^{-1}$ for $\omega \in {\pi }_1(A_{\alpha }\cap A_{\beta })$, and hence $\Phi$ induces an isomorphism ${\pi }_1(X)\cong {\ast }_{\alpha }{\pi }_1(A_{\alpha })/N$.

Example 1.40. Wedge sum: ${\pi }_1({\vee }_{\alpha }{X}_{\alpha })\cong {\ast }_{\alpha }{\pi }_1(X_{\alpha })$.

Remark 1.41. Van Kampen’s theorem is often applied when there are just two sets $A_{\alpha }$ and $A_{\beta }$ in the cover of $X$, so the condition on the triple intersections $A_{\alpha }\cap A_{\beta }\cap A_{\gamma }$ is superfluous and one obtains an isomorphism ${\pi }_1(X)\cong ({\pi }_1(A_{\alpha })\ast {\pi }_1(A_{\beta }))/N$, under the assumption that $A_{\alpha }\cap A_{\beta }$ is path-connected.

Example 1.42. Linking of Circles

 The complement ${\mathbb{R}}^3-A$ of a single circle $A$ deformation retracts onto a wedge sum $S^1\vee S^2$ embedded in ${\mathbb{R}}^3-A$. Hence ${\pi }_1({\mathbb{R}}^3-A)\cong {\pi }_1(S^1\vee S^2)\cong \mathbb{Z}$.

 In similar fashion, the complement ${\mathbb{R}}^3-(A\cup B)$ of two unlinked circles $A$ and $B$ deformation retracts onto $S^1\vee S^1\vee S^2\vee S^2$, hence, ${\pi }_1({\mathbb{R}}^3-(A\cup B))\cong {\pi }_1(S^1\vee S^1\vee S^2\vee S^2)=\mathbb{Z}\ast \mathbb{Z}$.

 On the other hand, if $A$ and $B$ are linked, then ${\mathbb{R}}^3-(A\cup B)$ deformation retracts onto the wedge sum of $S^2$ and a torus $S^1\times S^1$ separating $A$ and $B$, hence ${\pi }_1({\mathbb{R}}^3-(A\cup B))\cong {\pi }_1(S^1\times S^1)\cong \mathbb{Z}\times \mathbb{Z}$.

 For details and figures, please refer to Hatcher P46.

 For the remainder of this section we shall be interested in cell complexes, and in particular in how the fundamental group is affected by attaching $2$-cells.

 Suppose we attach a collection of $2$-cells $e_{\alpha }^2$ to a path-connected space $X$ via maps ${\phi }_{\alpha }:S^1\to X$, producing a space $Y$. If $s_0$ is a basepoint of $S^1$ then ${\phi }_{\alpha }$ determines a loop at ${\phi }_{\alpha }(s_0)$ that we shall call ${\phi }_{\alpha }$.

 For different $\alpha$’s the basepoint ${\phi }_{\alpha }(s_0)$ of these loops ${\phi }_{\alpha }$ may not all coincide. To remedy this, choose a basepoint $x_0\in X$ and a path ${\gamma }_{\alpha }$ in $X$ from $x_0$ to ${\phi }_{\alpha }(s_0)$ for each $\alpha$. Then ${\gamma }_{\alpha }{\phi }_{\alpha }\overline{{\gamma }_{\alpha }}$ is a loop at $x_0$.

 This loop may not be nullhomotopic in $X$, but it will certainly be nullhomotopic after the cell $e_{\alpha }^2$ is attached. Thus the normal subgroup $N\subset {\pi }_1(X,x_0)$ generated by all the loops ${\gamma }_{\alpha }{\phi }_{\alpha }\overline{{\gamma }_{\alpha }}$ for varying $\alpha$ lies in the kernel of the map ${\pi }_1(X,x_0)\to {\pi }_1(Y,x_0)$ induced by the inclusion $X\hookrightarrow Y$.

Proposition 1.43. If $Y$ is obtained from $X$ by attaching $2$-cells as described above, then the inclusion $X\hookrightarrow Y$ induces a surjection ${\pi }_1(X,x_0)\to {\pi }_1(Y,x_0)$ whose kernel is $N$. Thus ${\pi }_1(Y)\cong {\pi }_1(X)/N$.

Proposition 1.44. If $Y$ is obtained from $X$ by attaching $n$-cells for a fixed $n>2$, then the inclusion $X\hookrightarrow Y$ induces an isomorphism ${\pi }_1(X,x_0)\cong {\pi }_1(Y,x_0)$.

 For a path-connected cell complex $X$ the inclusion of the $2$-skeleton $X^2\hookrightarrow X$ induces an isomorphism ${\pi }_1(X^2,x_0)\cong {\pi }_1(X,x_0)$.

Remark 1.45. $N$ is independent of the choice of the paths ${\gamma }_{\alpha }$. If we replace ${\gamma }_{\alpha }$ by another path ${\eta }_{\alpha }$ having the same endpoints, then ${\gamma }_{\alpha }{\phi }_{\alpha }\overline{{\gamma }_{\alpha }}$ changes to ${\eta }_{\alpha }{\phi }_{\alpha }\overline{{\eta }_{\alpha }}=({\eta }_{\alpha }\overline{{\gamma }_{\alpha }}){\gamma }_{\alpha }{\phi }_{\alpha }\overline{{\gamma }_{\alpha }}({\gamma }_{\alpha }\overline{{\eta }_{\alpha }})$, so ${\gamma }_{\alpha }{\phi }_{\alpha }\overline{{\gamma }_{\alpha }}$ and ${\eta }_{\alpha }{\phi }_{\alpha }\overline{{\eta }_{\alpha }}$ define conjugate elements of ${\pi }_1(X,x_0)$.

Definition 1.46. $⟨g_{\alpha }\mid r_{\beta }⟩$ denotes the group with generators $g_{\alpha }$ and relator $r_{\beta }$, in other words, the free group on the generator $g_{\alpha }$ modulo the normal subgroup generated by the words $r_{\beta }$ in these generators.

Proposition 1.47. $${\pi }_1(M_g)\cong ⟨a_1,b_1,\cdots ,a_g,b_g\mid [a_1,b_1]\cdots [a_g,b_g]⟩.$$

Corollary 1.48. The surface $M_g$ is not homeomorphic, or even homotopy equivalent, to $M_h$ if $g\ne h$.

Proposition 1.49. $${\pi }_1(N_g)=⟨a_1,\cdots ,a_g\mid a_1^2\cdots a_g^2⟩.$$ This abelianizes to the direct sum of ${\mathbb{Z}}_2$ with $g-1$ copies of $\mathbb{Z}$ since in the abelianzation we can rechoose the generators to be $a_1,\cdots ,a_{g-1}$ and $a_1+\cdots +a_g$ with $2(a_1+\cdots +a_g)=0$.

Corollary 1.50. $N_g$ is not homotopy equivalent to $N_h$ if $g\ne h$, nor is $N_g$ homotopy equivalent to any orientable surface $M_h$.

Corollary 1.51. For every group $G$, there is a $2$-dimensional cell complex $X_G$ with ${\pi }_1(X_G)\cong G$.

Remark 1.52. However, the cell complex $X_G$ may not be a surface. For more details, please refer to Hatcher P52.

Covering Spaces

 The connection between the fundamental group and covering spaces runs much deeper than this, and in many ways they can be regarded as two viewpoints toward the same thing. Algebraic aspects of the fundamental group can often be translated into geometric language of covering spaces.

Definition 1.53. A covering space of a space $X$ is a space $\tilde{X}$ together with a map $p:\tilde{X}\to X$ satisfying the following condition: Each point $x\in X$ has an open neighborhood $U$ in $X$ such that $p^{-1}(U)$ is a union of disjoint open sets in $\tilde{X}$, each of which is mapped homeomorphically onto $U$ by $p$.

 Such a $U$ is called evenly covered and the disjoint open sets in $\tilde{X}$ that project homeomorphically to $U$ by $p$ are called sheets of $\tilde{X}$ over $U$.

 We allow $p^{-1}(U)$ to be empty, so $p$ need not be surjective. Remark that this definition varies from person to person.

 The number of sheets over $U$ is the cardinality of $p^{-1}(x)$ for $x\in U$. As $x$ varies over $X$ this number is locally constant, so it is constant if $X$ is connected.

Definition 1.54. A lift of a map $f:Y\to X$ is a map $\tilde{f}:Y\to \tilde{X}$ such that $p\tilde{f}=f$.

Proposition 1.55. Homotopy lifting property

 Given a covering space $p:\tilde{X}\to X$, a homotopy $f_t:Y\to X$, and a map $\widetilde{f_0}:Y\to \tilde{X}$ lifting $f_0$, then there exists a unique homotopy $\widetilde{f_t}:Y\to \tilde{X}$ of $\widetilde{f_0}$ that lifts $f_t$.

Remark 1.56. Taking $Y$ to be a point gives the path lifting property.

Proposition 1.57. The map $p_{\ast }:{\pi }_1(\tilde{X},\widetilde{x_0})\to {\pi }_1(X,x_0)$ induced by a covering space $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ is injective.

 The image subgroup $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$ in ${\pi }_1(X,x_0)$ consists of the homotopy classes of loops in $X$ based at $x_0$ whose lift to $\tilde{X}$ starting at $\widetilde{x_0}$ are loops.

Proposition 1.58. The number of sheets of a covering space $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ with $X$ and $\tilde{X}$ path-connected equals the index of $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$ in ${\pi }_1(X,x_0)$.

Proposition 1.59. Lifting criterion

 Suppose given a covering space $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ and a map $f:(Y,y_0)\to (X,x_0)$ with $Y$ path-connected and locally path-connected.

 Then a lift $\tilde{f}:(Y,y_0)\to (\tilde{X},\widetilde{x_0})$ of $f$ exists iff $f_{\ast }({\pi }_1(Y,y_0))\subset p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$.

Proposition 1.60. Unique lifting property

 Given a covering sapce $p:\tilde{X}\to X$ and a map $f:Y\to X$, if two lifts $\widetilde{f_1},\widetilde{f_2}:Y\to \tilde{X}$ of $f$ agree at one point of $Y$ and $Y$ is connected, then $\widetilde{f_1}$ and $\widetilde{f_2}$ agree on all of $Y$.

The Classification of Covering spaces

 The main thrust of the classification will be a correspondence between connected covering spaces of $X$ and subgroups of ${\pi }_1(X)$. This is often called the Galois correspondence because of its surprising similarity to another basic correspondence in the purely algebraic subject of Galois theory.

 The Galois correspondence arises from the function that assigns to each covering space $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ the subgroup $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$ of ${\pi }_1(X,x_0)$.  Fisrt we consider whether this function is surjective. That is, we ask whether every subgroup of ${\pi }_1(X,x_0)$ is realized as $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$ for some covering space $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$. In particular we can ask whether the trivial subgroup is realized. Since $p_{\ast }$ is always injective, this amounts to asking whether $X$ has a simply-connected covering space.

Definition 1.61. Semilocally simply-connected

 We say $X$ is semilocally simply-connected if each point $x\in X$ has a neighborhood $U$ such that the inclusion-induced map ${\pi }_1(U,x)\to {\pi }_1(X,x)$ is trivial.

 A locally simply-connected space is certainly semilocally simply-connected. An example of a space that is not semilocally simply-connected is the shrinking wedge of circles.

 Given a path-connected, locally path-connected, semilocally simply-connected space $X$ with a basepoint $x_0\in X$, we are therefore led to define $$\tilde{X}=\{[\gamma ]\mid \gamma \text{is a path in}X\text{starting at}{x}_0\}.$$ Let $\mathcal{U}$ be the collection of path-connected open sets $U\subset X$ such that ${\pi }_1(U)\to {\pi }_1(X)$ is trivial. Say that $\mathcal{U}$ is a basis for the topology on $X$ if $X$ is locally path-connected, semilocally simply-connected.

 Given a set $U\in \mathcal{U}$ and a path $\gamma$ in $X$ from $x_0$ to a point in $U$, let $$U_{[\gamma ]}=\{[\gamma \cdot \eta ]\mid \eta \text{is a path in}U\text{with}\eta (0)=\gamma (1)\}.$$$U_{[\gamma ]}$ depends only on the homotopy class $[\gamma ]$ and $U_{[\gamma ]}=U_{[{\gamma }^{\prime }]}$ if $[{\gamma }^{\prime }]\in U_{[\gamma ]}$. And the sets $U_{[\gamma ]}$ form a basis for a topology basis for a topology on $\tilde{X}$.

 The bijection $p:U_{[\gamma ]}\to U$ is a homeomorphism. This implies $p:\tilde{X}\to X$ is continuous and then we can also deduce that this is a covering space.

 It remains only to show that $\tilde{X}$ is simply-connected. Then we complete the construction of a simply-connected covering space $\tilde{X}\to X$.

 The hypotheses for constructing a simply-connected covering space of $X$ in fact suffice for constructing covering spaces realizing arbitrary subgroup of ${\pi }_1(X)$.

Proposition 1.62. Suppose $X$ is path-connected, locally path-connected, semilocally simply-connected. Then for every subgroup $H\subset {\pi }_1(X,x_0)$ there is a covering space $p:X_H\to X$ such that $p_{\ast }({\pi }_1(X_H,\widetilde{x_0}))=H$ for a suitably chosen basepoint $\widetilde{x_0}\in X_H$.

Definition 1.63. An isomorphism between covering spaces $p_1:\widetilde{X_1}\to X$ and $p_2:\widetilde{X_2}\to X$ is a homeomorphism $f:\widetilde{X_1}\to \widetilde{X_2}$ such that $p_1=p_{2}f$.

Proposition 1.64. If $X$ is a path-connected and locally path-connected, then two path-connected covering spaces $p_1:\widetilde{X_1}\to X$ and $p_2:\widetilde{X_2}\to X$ are isomorphic via an isomorphism $f:\widetilde{X_1}\to \widetilde{X_2}$ taking a basepoint $\widetilde{x_1}\in p_1^{-1}(x_0)$ to a basepoint $\widetilde{x_2}\in p_2^{-1}(x_0)$ iff $p_{1\ast }({\pi }_1(\widetilde{X_1},\widetilde{x_1}))=p_{2\ast }({\pi }_1(\widetilde{X_2},\widetilde{x_2}))$.

Theorem 1.65. Classification Theorem

 Let $X$ be a path-connected, locally path-connected, semilocally simply-connected space. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ and the set of subgroups of ${\pi }_1(X,x_0)$, obtained by associating the subgroup $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))$ to the covering space $(\tilde{X},\widetilde{x_0})$.

 If basepoints are ignored, this correspondence gives a bijection between isomorphism classes of path-connected covering sapce $p:\tilde{X}\to X$ and conjugacy classes of subgroups of ${\pi }_1(X,x_0)$.

Definition 1.66. A simply-connected covering space of a path-connected, locally path-connected space $X$ is a covering sapce of every other path-connected covering space of $X$.

 A simply-connected covering space of $X$ is therefore called a universal cover. It is unique up to isomorphism, so one is justified in calling it the universal covering.

Definition 1.67. For a covering space $p:\tilde{X}\to X$ the isomorphisms $\tilde{X}\to \tilde{X}$ are called deck transformations or covering transformations. These form a group $G(\tilde{X})$ under composition.

Example 1.68. $p:\mathbb{R}\to S^1$ projecting a vertical helix onto a circle, $G(\tilde{X})=\mathbb{Z}$ in this case.

Example 1.69. The $n$-sheeted covering space $p:S^1\to S^1,z\mapsto z^n$, the deck transformations are the rotations of $S^1$ through angles that the multiples of $2\pi /n$, so $G(\tilde{X})={\mathbb{Z}}_n$.

 By the unique lifting property, a deck transformation is completely determined by where it sends a single point, assuming $\tilde{X}$ is path-connected. In particular, only the identity deck transformation can fix a point of $\tilde{X}$.

Definition 1.70. A covering space $p:\tilde{X}\to X$ is called normal if for each $x\in X$ and each pair of lifts $\tilde{x},{\tilde{x}}^{\prime }$ of $x$ there is a deck transformation taking $\tilde{x}$ to ${\tilde{x}}^{\prime }$.

 For example, the covering space $\mathbb{R}\to S^1$ and the $n$-sheeted covering spaces $S^1\to S^1$ are normal.

Proposition 1.71. Let $p:(\tilde{X},\widetilde{x_0})\to (X,x_0)$ be a path-connected covering space of the path-connected, locally path-connected space $X$, ane let $H$ be the subgroup $p_{\ast }({\pi }_1(\tilde{X},\widetilde{x_0}))\subset {\pi }_1(X,x_0)$. Then:

  (a) This covering space is normal iff $H$ is a normal subgroup of ${\pi }_1(X,x_0)$.

  (b) $G(\tilde{X})$ is isomorphic to the quotient $N(H)/H$ where $N(H)$ is the normalizer of $H$ in ${\pi }_1(X,x_0)$.

 In particular, $G(\tilde{X})$ is isomorphic to ${\pi }_1(X,x_0)/H$ if $\tilde{X}$ is a normal covering. Hence for the universal cover $\tilde{X}\to X$ we have $G(\tilde{X})\cong {\pi }_1(X)$.

Definition 1.72. We call an action satisfying the following condition is a covering space action or properly discontinuous action.

 Each $y\in Y$ has a neighborhood $U$ such that all the images $g(U)$ for varying $g\in G$ are disjoint. In other words, $g_1(U)\cap g_2(U)\ne \varnothing$ implies $g_1=g_2$.

 Equivalently, $U\cap g(U)\ne \varnothing$ only when $g$ is the identity.

Proposition 1.73. The action of the deck transformation group $G(\tilde{X})$ on $\tilde{X}$ is a covering space action.

Definition 1.74. The orbits $Gy=\{g(y)\mid g\in G\}$ in $Y$, and $Y/G$ is called the orbit space of the action.

 For example, for a normal covering space $\tilde{X}\to X$, the orbit space $\tilde{X}/G(\tilde{X})$ is just $X$.

Proposition 1.75. If an action of a group $G$ on a space $Y$ satisfies covering space action condition, then

 (a) The quotient map $p:Y\to Y/G$, $p(y)=Gy$, is a normal covering space.

 (b) $G$ is the group of deck transformation of this covering space $Y\to Y/G$ if $Y$ is path-connected.

 (c) $G$ is isomorphic to ${\pi }_1(Y/G)/p_{\ast }({\pi }_1(Y))$ if $Y$ is path-connected and locally path-connected.

Corollary 1.76. For a covering space action of a group $G$ on a simply-connected locally path-connected space $Y$, the orbit space $Y/G$ has fundamental group isomorphic to $G$.

Example 1.77. A covering sapce $M_{mn+1}\to M_{m+1}$. For details please refer to Hatcher P73.

Example 1.78. $\mathbb{R}{P}^n$.

 The antipodal map of $S^n$, $x\mapsto -x$, generates an action of ${\mathbb{Z}}_2$ on $S^n$ with orbit space $\mathbb{R}{P}^n$, real projective $n$-space.

 Since $S^n$ is simply-connected if $n\ge 2$, we deduce that ${\pi }_1(\mathbb{R}{P}^n)\cong {\mathbb{Z}}_2$ for $n\ge 2$.

2Homology

Simplicial and Singular Homology

Simplicial homology

Example 2.1. $X=S^1$ with one vertex $v$ and one edge $e$. Then ${\Delta }_0(S^1)$ and ${\Delta }_1(S^1)$ are both $\mathbb{Z}$ and the boundary map ${\partial }_1$ is zero since $\partial e=v-v$. Hence $$H_n^{\Delta }(S^1)\cong \{\begin{array}{ll}\mathbb{Z},& n=0,1\\ 0,& n\ge 2\end{array}.$$

Example 2.2. $X=T$, the torus with the $\Delta$-complex structure (a square identifying the opposite edges $a,b$ and we draw one diagonal denoted by $c$.) Denote the $2$-simplices $U$ and $L$.

 ${\partial }_1=0$ so $H_1^{\Delta }(T)\cong \mathbb{Z}$. Since ${\partial }_{2}U=a+b-c={\partial }_{2}L$ and $\{a,b,a+b-c\}$ is a basis for ${\Delta }_1(T)$, it follows that $H_1^{\Delta }(T)\cong \mathbb{Z}\oplus \mathbb{Z}$ with basis the homology classes $[a]$ and $[b]$.

 Since there are no $3$-complices, $H_2^{\Delta }(T)$ is equal to $\mathrm{Ker}{\partial }_2$, which is infinite cyclic generated by $U-L$. Thus $$H_n^{\Delta }(T)\cong \{\begin{array}{ll}\mathbb{Z}\oplus \mathbb{Z},& n=1\\ \mathbb{Z},& n=0,2\\ 0,& n\ge 3\end{array}.$$

Example 2.3. $X=\mathbb{R}{P}^2$, the projective plane with the $\Delta$-complex structure (a square identifying the opposite edges $a,b$ with orientation revered and we draw one diagonal denoted by $c$.) Denote the $0$-simplices $w$ and $v$, the $2$-simplices $U$ and $L$.

 The image of ${\partial }_1$ is generated by $w-v$, so $H_0^{\Delta }(X)\cong \mathbb{Z}$ with either vertex as a generator.

 Since ${\partial }_{2}U=-a+b+c$ and ${\partial }_{2}L=a-b+c$, we see that ${\partial }_2$ is injective, so $H_2^{\Delta }(X)=0$.

 Further, $\mathrm{Ker}{\partial }_1\cong \mathbb{Z}\oplus \mathbb{Z}$ with basis $a-b$ and $c$ and $\mathrm{Im}{\partial }_2$ is an index-two subgroup of $\mathrm{Ker}{\partial }_1$ since we can choose $c$ and $a-b+c$ as a basis for $\mathrm{Ker}{\partial }_1$ and $a-b+c$ and $2c=(a-b+c)+(-a+b+c)$ as a basis of $\mathrm{Im}{\partial }_2$. Thus $H_1^{\Delta }(X)\cong {\mathbb{Z}}_2$. $$H_n^{\Delta }(\mathbb{R}{P}^2)\cong \{\begin{array}{ll}{\mathbb{Z}}_2,& n=1\\ \mathbb{Z},& n=0\\ 0,& n\ge 2\end{array}.$$

Example 2.4. We can obtain a $\Delta$-complex structure on $S^n$ by taking two copies of ${\Delta }^n$ and identifying their boundaries via the identity map. Labeling these two $n$-simplices $U$ and $L$, then it is obvious that $\mathrm{Ker}{\partial }_N$ is infinite cyclic generated by $U-L$. Thus $H_n^{\Delta }(S^n)\cong \mathbb{Z}$ for this $\Delta$-complex structure on $S^n$.

Singular homology

Definition 2.5. A singular $n$-simplex in a space $X$ is by definition just a continuous map $\sigma :{\Delta }^n\to X$.

Remark 2.6. The word “singular” is used here to express the idea that $\sigma$ need not be a nice embedding but can have “singularities” where its image does not look at all like a simplex.

Definition 2.7. Let $C_n(X)$ be the free abelian group with basis the set of singular $n$-simplices in $X$. Elements of $C_n(X)$, called $n$-chains, are finite formal sums ${\sum }_i{n}_i{\sigma }_i$ for $n_i\in \mathbb{C}$ and ${\sigma }_i:{\Delta }^n\to X$.

Definition 2.8. Singular homology group $H_n(X)=\mathrm{Ker}{\partial }_n/\mathrm{Im}{\partial }_{n+1}$.

Remark 2.9. It is evident from the definition that homeomorphic spaces have isomorphic singular homology groups $H_n$, in contrast with the situation for $H_n^{\Delta }$.

Proposition 2.10. Corresponding to the decomposition of a space $X$ into its path-components $X_{\alpha }$ there is an isomorphism of $H_n(X)$ with the direct sum ${⨁}_{\alpha }{H}_n(X_{\alpha })$.

Proposition 2.11. If $X$ is nonempty and path-connected, then $H_0(X)\cong \mathbb{Z}$. Hence for any space $X$, $H_0(X)$ is a direct sum of $\mathbb{Z}$’s, one for each path-component of $X$.

Proposition 2.12. If $X$ is a point, then $H_n(X)=0$ for $n>0$ and $H_0(X)\cong \mathbb{Z}$.

Definition 2.13. Define the reduced homology groups ${\tilde{H}}_n(X)$ to be the homology groups of the augmented chain complex $$\cdots \to C_2(X)\stackrel{{\partial }_2}{\to }{C}_1(X)\stackrel{{\partial }_1}{\to }{C}_0(X)\stackrel{\epsilon }{\to }\mathbb{Z}\to 0$$where $\epsilon ({\sum }_i{n}_i{\sigma }_i)={\sum }_i{n}_i$.

Remark 2.14. Formally, one can think of the extra $\mathbb{Z}$ in the augmented chain complex as generated by the unique map $[\varnothing ]\to X$ where $[\varnothing ]$ is the empty simplex, with no vertices. The augmentation map $\epsilon$ is then the usual boundary map since $\partial [v_0]=[{\hat{v}}_0]=[\varnothing ]$.

Proposition 2.15. A chain map between chain complexes induces homomorphisms between the homology groups of the two complexes.

Theorem 2.16. If two maps $f,g:X\to Y$ are homotopic, then they induce the same homomorphism $f_{\ast }=g_{\ast }:H_n(X)\to H_n(Y)$.

Corollary 2.17. The maps $f_{\ast }:H_n(X)\to H_n(Y)$ induced by a homotopy equivalence $f:X\to Y$ are isomorphisms for all $n$.

Proposition 2.18. Chain-homotopic chain maps induce the same homomorphism on homology.

 The properties of induced homomorphisms we stated above hold equally well in the setting of reduced homology, with the same proofs.

Exact Sequences and Excision

 If there was always a simple relationship between the homology groups of a space $X$, a subspace $A$, and the quotient space $X/A$, then this could be a very useful tool in understanding the homology groups of spaces such as CW complexes that can be built inductively from successively more complicated subspaces.

 The novel feature of the actual relationship is that it involves the groups $H_n(X)$, $H_n(A)$ and $H_n(X/A)$ for all values of $n$ simultaneously.

 In practice this is not as bad as it might sound, and in addition it has the pleasant side effect of sometimes allowing high-dimensional homology groups to be computed in terms of lower-dimensional groups which may already be known, for example by induction.

Theorem 2.19. If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $$\cdots \to {\tilde{H}}_n(A)\stackrel{i_{\ast }}{\to }{\tilde{H}}_n(X)\stackrel{j_{\ast }}{\to }{\tilde{H}}_n(X/A)\stackrel{\partial }{\to }{\tilde{H}}_{n-1}(A)\stackrel{i_{\ast }}{\to }{\tilde{H}}_{n-1}(X)\to \cdots \to {\tilde{H}}_0(X/A)\to 0$$where $i$ is the inclusion $X\hookrightarrow X$ and $j$ is the quotient map $X\to X/A$.

Remark 2.20. The map $\partial$ will be constructed in the course of the proof. The idea is that an element $x\in {\tilde{H}}_n(X/A)$ can be represented by a chain $\alpha$ in $X$ with $\partial \alpha$ a cycle in $A$ whose homology class is $\partial x\in {\tilde{H}}_{n-1}(A)$.

Definition 2.21. Pairs of spaces $(X,A)$ satisfying the hypothesis of the theorem will be called good pairs.

 For example, if $X$ is a CW complex and $A$ is a nonempty subcomplex, then $(X,A)$ is a good pair.

Corollary 2.22. ${\tilde{H}}_n(S^n)\cong \mathbb{Z}$ and ${\tilde{H}}_i(S^n)=0$ for $i\ne n$.

Proof.

Proof. For $n>0$, take $(X,A)=(D^n,S^{n-1})$ so $X/A=S^n$. Then we could show that ${\tilde{H}}_i(S^n)\cong {\tilde{H}}_{i-1}(S^{n-1})$.

$\square$

Corollary 2.23. Brouwer theorem

 $\partial D^n$ is not a retract of $D^n$. Hence every map $f:D^n\to D^n$ has a fixed point.

Definition 2.24. Let $C_n(X,A)$ be the quotient group $C_n(X)/C_n(A)$. The boundary map $\partial$ is induced by the $\partial$ in $C_n(X)$.

 ${\partial }^2=0$ holds then $C_{\ast }(X,A)$ becomes a complex, the homology group $H_n(X,A)$ corresponding to this complex is called relative homology groups.

 Elements in $Z_n(X,A)$ are called relative cycles, $n$-chains $\alpha \in C_n(X)$ with $\partial \alpha \in C_{n-1}(A)$.

 Elements in $B_n(X,A)$ are called relative boundaries, $\alpha =\partial \beta +\gamma$ for $\beta \in C_{n+1}(X)$ and $\gamma \in C_n(A)$.

Proposition 2.25. The commutative diagram with the columns are exact and the rows are chain complexes which we denote $A$, $B$ and $C$. Such a diagram is called a short exact sequence of chain complexes. This short exact sequence of chain complexes stretches out into a long exact sequence of homology groups $$\cdots H_n(A)\stackrel{i_{\ast }}{\to }{H}_n(B)\stackrel{j_{\ast }}{\to }{H}_n(C)\stackrel{\partial }{\to }{H}_{n-1}(A)\stackrel{i_{\ast }}{\to }{H}_{n-1}(B)\to \cdots$$where $H_n(A)$ denotes the homology group $\mathrm{Ker}\partial /\mathrm{Im}\partial$ at $A_n$ in the chain complex $A$, and $H_n(B)$ and $H_n(C)$ are defined similarly.

Now define the boundary map $\partial :H_n(C)\to H_{n-1}(A)$. Let $c\in C_n$ be a cycle, and $c=jb$ for some $b\in B_n$ as $j$ is onto. Since $j\partial b=\partial c=0$ implies $\partial b\in \mathrm{Ker}j=\mathrm{Im}i$, we have $\partial b=ia$ for some $a\in A_{n-1}$, where $\partial a=0$ since $i\partial a=\partial \partial b=0$ and $i$ is injective. Take $\partial [c]=[a]$.  

The theorem represents the beginnings of the subject of homological algebra. The method of proof is sometimes called diagram chasing.

Proposition 2.26. From the last proposition and the diagram we have the following exact sequence $$\cdots \to H_n(A)\stackrel{i_{\ast }}{\to }{H}_n(X)\stackrel{j_{\ast }}{\to }{H}_n(X,A)\stackrel{\partial }{\to }{H}_{n-1}(A)\stackrel{i_{\ast }}{\to }{H}_{n-1}(X)\to \cdots .$$

Proposition 2.27. There is a completely analogous long exact sequence of reduced homology groups for a pair $(X,A)$ with $A\ne \varnothing$.

 In particular, ${\tilde{H}}_n(X,A)$ is the same as $H_n(X,A)$ for all $n$, where $A\ne \varnothing$.

Example 2.28. $$H_i(D^n,\partial D^n)\cong \{\begin{array}{ll}\mathbb{Z},& i=n\\ 0,& \text{otherwise}\end{array}.$$

Example 2.29. $H_n(X,x_0)\cong {\tilde{H}}_n(X)$ for all $n$.

Proposition 2.30. If two maps $f,g:(X,A)\to (Y,B)$ are homotopic through maps of pairs $(X,A)\to (Y,B)$, then $f_{\ast }=g_{\ast }:H_n(X,A)\to H_n(Y,B)$.

Proposition 2.31. An easy generalization of the long exact sequence of a pair $(X,A)$ is the long exact sequence of a triple $(X,A,B)$, where $B\subset A\subset X$: $$\cdots \to H_n(A,B)\to H_n(X,B)\to H_n(X,A)\to H_{n-1}(A,B)\to \cdots .$$

Theorem 2.32. Excision Theorem

 Given subspaces $Z\subset A\subset X$ such that $\overline{Z}\subset \circ A$, then the inclusion $(X-Z,A-Z)\hookrightarrow (X,A)$ induces isomorphisms $H_n(X-Z,A-Z)\to H_n(X,A)$ for all $n$.

 Equivalently, for subspaces $A,B\subset X$ whose interiors cover $X$, the inclusion $(B,A\cap B)\hookrightarrow (X,A)$ induces isomorphism $H_n(B,A\cap B)\to H_n(X,A)$ for all $n$.

Remark 2.33. In a metric space “smallness” can be defined in terms of diameters, but for general spaces it will be defined in terms of covers.

Definition 2.34. Let $C_n^{\mathcal{U}}(X)$ be the subgroup of $C_n(X)$ consisting of chains ${\sum }_i{n}_i{\sigma }_i$ such that each ${\sigma }_i$ has image contained in some set in the cover $\mathcal{U}$.

Proposition 2.35. The inclusion $\iota :C_n^{\mathcal{U}}(X)\hookrightarrow C_n(X)$ is a chain homotopy equivalence, that is, there is a chain map $\rho :C_n(X)\to C_n^{\mathcal{U}}(X)$ such that $\iota \rho$ and $\rho \iota$ are chain homotopic to the identity.

 Hence $\iota$ induces isomorphisms $H_n^{\mathcal{U}}(X)\cong H_n(X)$ for all $n$.

Proposition 2.36. For good pairs $(X,A)$, the quotient map $q:(X,A)\to (X/A,A/A)$ induces isomorphisms $q_{\ast }:H_n(X,A)\to H_n(X/A,A/A)\cong {\tilde{H}}_n(X/A)$ for all $n$.

Remark 2.37. This proposition shows that relative homology can be expressed as reduced absolute homology in the case of good pairs $(X,A)$, but in fact there is a way of doing this for arbitrary pairs: $${\tilde{H}}_n(X\cup CA)\cong H_n(X\cup CA,CA)\cong H_n(X\cup CA-\{p\},CA-\{p\})\cong H_n(X,A),$$where $p\in CA$ is the tip of the cone.

Example 2.38. The identity map $i_n:{\Delta }^n\to {\Delta }^n$, viewed as a singular $n$-simplex, is a cycle generating $H_n({\Delta }^n,\partial {\Delta }^n)$. We claim there are isomorphisms $$H_n(D^n,\partial D^n)\cong H_n({\Delta }^n,\partial {\Delta }^n)\stackrel{\cong }{\to }{H}_{n-1}(\partial {\Delta }^n,\Lambda )\stackrel{\cong }{\leftarrow }{H}_{n-1}({\Delta }^{n-1},\partial {\Delta }^{n-1}),$$where $\Lambda \subset {\Delta }^n$ is the union of all but one of the $(n-1)$-dimensional faces of ${\Delta }^n$.

Example 2.39. To find a cycle generating ${\tilde{H}}_n(S^n)$ let us regard $S^n$ as two $n$-simplices ${\Delta }_1^n$ and ${\Delta }_2^n$ with their boundaries identified in the obvious way, preserving the ordering of vertices.

 The difference ${\Delta }_1^n-{\Delta }_2^n$, viewed as a singular $n$-chain, is then a cycle, and we claim it represents a generator of ${\tilde{H}}_n(S^n)$. To see this, consider the isomorphisms $${\tilde{H}}_n(S^n)\stackrel{\cong }{\to }{H}_n(S^n,{\Delta }_2^n)\stackrel{\cong }{\leftarrow }{H}_n({\Delta }_1^n,\partial {\Delta }_1^n).$$

Corollary 2.40. If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A\cap B)\hookrightarrow (X,A)$ induces isomorphisms $H_n(B,A\cap B)\to H_n(X,A)$ for all $n$.

Corollary 2.41. For a wedge sum ${\bigvee }_{\alpha }{X}_{\alpha }$, the inclusions $i_{\alpha }:X_{\alpha }\hookrightarrow {\bigvee }_{\alpha }{X}_{\alpha }$ induce an isomorphism ${⨁}_{\alpha }{i}_{{\alpha }_{\ast }}:{⨁}_{\alpha }{\tilde{H}}_n(X_{\alpha })\to {\tilde{H}}_n({\bigvee }_{\alpha }{X}_{\alpha })$, provided that the wedge sum is formed at basepoints $x_{\alpha }\in X_{\alpha }$ such that the pairs $(X_{\alpha },x_{\alpha })$ are good.

Proof.

Proof. Since reduced homology is the same as homology relative to a basepoint, this follows from the proposition by taking $(X,A)=({⨆}_{\alpha }{X}_{\alpha },{⨆}_{\alpha }\{x_{\alpha }\})$.

$\square$

Theorem 2.42. If nonempty open sets $U\subset {\mathbb{R}}^m$ and $V\subset {\mathbb{R}}^n$ are homeomorphic, then $m=n$.

Proof.

Proof. For $x\in U$ we have $H_k(U,U-\{x\})\cong H_k({\mathbb{R}}^m,{\mathbb{R}}^m-\{x\})\cong {\tilde{H}}_{k-1}({\mathbb{R}}^m-\{x\})\cong {\tilde{H}}_{k-1}(S^{m-1})$ by excision, the long exact sequence for $({\mathbb{R}}^m,{\mathbb{R}}^m-\{x\})$, and deformation retraction respectively.

$\square$

Definition 2.43. The local homology groups of a space $X$ at a point $x\in X$ are defined to be the groups $H_n(X,X-\{x\})$.

 For any open neighborhood $U$ of $x$, excision gives isomorphisms $H_n(X,X-\{x\})\cong H_n(U,U-\{x\})$ assuming points are closed in $X$, and thus the groups $H_n(X,X-\{x\})$ depend only on the local topology of $X$ near $x$.

 A homeomorphism $f:X\to Y$ must induce isomorphisms $H_n(X,X-\{x\})\cong H_n(Y,Y-\{f(x)\})$ for all $x$ and $n$, so the local homology groups can be used to tell when spaces are not locally homeomorphic at certain points.

Proposition 2.44. Naturality

 For six chain complexes $A,B,C,A^{\prime },B^{\prime },C^{\prime }$ with $$0\to A_n\stackrel{i}{\to }{B}_n\stackrel{j}{\to }{C}_n\to 0$$$$0\to A_n^{\prime }\stackrel{i^{\prime }}{\to }{B}_n^{\prime }\stackrel{j^{\prime }}{\to }{C}_n^{\prime }\to 0$$for any $n$. Moreover, there are three chain maps $\alpha :A\to A^{\prime }$, $\beta :B\to B^{\prime }$, $\gamma :C\to C^{\prime }$. Then the induced diagram is commutative.

Proposition 2.45. For $f:(X,A)\to (Y,B)$, the diagram is commutative.

Proposition 2.46. Recall Theorem 2.19, if $X,Y$ are spaces and $A,B$ are nonempty closed subspaces that are deformation retracts of some neighborhoods in $X,Y$ respectively, then the following diagram commutes

The Equivalence of Simplicial and Singular Homology

Theorem 2.47. The homomorphisms $H_n^{\Delta }(X,A)\to H_n(X,A)$ are isomorphisms for all $n$ and all $\Delta$-complex pairs $(X,A)$.

Lemma 2.48. The Five-Lemma

 In a commutative diagram of abelian groups as shown, if the two rows are exact and $\alpha ,\beta ,\delta$ and $\epsilon$ are isomorphisms, then $\gamma$ is an isomorphism also.

Corollary 2.49. We can deduce from this theorem that $H_n(X)$ is finitely generated whenever $X$ is a $\Delta$-complex with finitely many $n$-simplices, since in this case the simplicial chain group ${\Delta }_n(X)$ is finitely generated, hence also its subgroup of cycles and therefore also the latter group’s quotient $H_n^{\Delta }(X)$.

Definition 2.50. If we write $H_n(X)$ as the direct sum of cyclic groups, then the number of $\mathbb{Z}$ summands is known traditionally as the $n^{\text{th}}$ Betti number of $X$, and integers specifying the orders of the finite cyclic summands are called torsion coefficients.

Degree

Definition 2.51. For a map $f:S^n\to S^n$ with $n>0$, the induced map $f_{\ast }:H_n(S^n)\to H_n(S^n)$ is a homomorphism from an infinite cyclic group to itself and so must be of the form $f_{\ast }\alpha =d\alpha$ for some integer $d$ depending only on $f$. This integer is called the degree of $f$, with the notation $\mathrm{deg}f$.

Proposition 2.52. Here are some basic properties of degree:

(a)	$\mathrm{deg}\mathrm{id}=1$.
(b)	$\mathrm{deg}f=0$ if $f$ is not surjective. (Hint: $f:S^n\to S^n-\{x\}\hookrightarrow S^n$.)
(c)	If $f\sim g$ then $\mathrm{deg}f=\mathrm{deg}g$ since $f_{\ast }=g_{\ast }$. The converse statement, that $f\sim g$ if $\mathrm{deg}f=\mathrm{deg}g$, is a fundamental theorem of Hopf from around 1925.
(d)	$\mathrm{deg}fg=\mathrm{deg}f\cdot \mathrm{deg}g$ since $(fg{)}_{\ast }=f_{\ast }{g}_{\ast }$. As a consequence, $\mathrm{deg}f=\pm 1$ if $f$ is a homotopy equivalence.
(e)	$\mathrm{deg}f=-1$ if $f$ is a reflection of $S^n$.
(f)	$\mathrm{deg}-\mathrm{id}=(-1{)}^{n+1}$ where $-\mathrm{id}$ is the antipodal map which is the composition of $n+1$ reflections.
(g)	If $f:S^n\to S^n$ has no fixed points, then $\mathrm{deg}f=(-1{)}^{n+1}$ since it is not difficult to prove $f\sim -\mathrm{id}$.

Theorem 2.53. $S^n$ has a continuous field of nonzero tangent vectors iff $n$ is odd.

Proof.

Proof. Suppose that the tangent vectors $v(x)$, orthogonal to $x$, are all of length $1$. The homotopy $f_t:S^n\to S^n,x\mapsto (\cos t)x+(\sin t)v(x)$ between $\mathrm{id}$ and $-\mathrm{id}$ implies $1=(-1{)}^{n+1}$.

$\square$

Proposition 2.54. ${\mathbb{Z}}_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even.

Proof.

Proof. Since homeomorphisms have degree $\pm 1$, an action of a group $G$ on $S^n$ determines a degree function $d:G\to \{\pm 1\}$. This is a homomorphism since $\mathrm{deg}fg=\mathrm{deg}f\cdot \mathrm{deg}g$. If the action is free, $d$ sends each nontrivial element of $G$ to $(-1{)}^{n+1}$ by property (g) above. Thus, when $n$ is even, $d$ has trivial kernel, so $G\subset {\mathbb{Z}}_2$.

$\square$

Cellular Homology

Lemma 2.55. If $X$ is a CW complex, then:

(a)	$H_k(X^n,X^{n-1})$ is zero for $k\ne n$ and is free abelian for $k=n$, with a basis in one-to-one correspondence with the $n$-cells of $X$.
(b)	$H_k(X^n)=0$ for $k>n$. In particular, if $X$ is finite-dimensional then $H_k(X)=0$ for $k>\dim X$.
(c)	The map $H_k(X^n)\to H_k(X)$ induced by the inclusion $X^n\hookrightarrow X$ is an isomorphism for $k<n$ and surjective for $k=n$.

Proof.

Proof.

(a)

$H_k(X^n,X^{n-1})={\tilde{H}}_k(X^n/X^{n-1})={\tilde{H}}_k(\bigvee S^n)=⨁{\tilde{H}}_k(S^n)$.

(b,c)

Consider $$H_{k+1}(X^n,X^{n-1})\to H_k(X^{n-1})\to H_k(X^n)\to H_k(X^n,X^{n-1})$$If $k\ne n$, the last term is $0$ so $H_k(X^{n-1})\to H_k(X^n)$ is surjective; if $k\ne n,n-1$, both the first term and last term are $0$ so $H_k(X^{n-1})\cong H_k(X^n)$. Therefore $H_k(X^n)\cong H_k(X^{n-1})\cong \cdots \cong H_k(X^0)=0$ for $k>n$, $H_k(X^n)\cong H_k(X^{n+1})\cong \cdots \cong H_k(X)$ for $k<n$, and $H_k(X^n)\to H_k(X^{n+1})\cong H_k(X)$ is surjective for $k=n$.

$\square$

Proposition 2.56. Let $X$ be a CW compelx. Using 2.55, portions of the long exact sequence for the pairs $(X^{n+1},X^n)$, $(X^n,X^{n-1})$ and $(X^{n-1},X^{n-2})$ fit into a diagram where $d_{n+1}$ and $d_n$ are defined as the compositions $j_n{\partial }_{n+1}$ and $j_{n-1}{\partial }_n$, which are just ‘relativizations’ of the boundary maps ${\partial }_{n+1}$ and ${\partial }_n$.

Definition 2.57. Note $d_{n+1}\circ d_n=0$, the horizontal row in the diagram is a chain complex, called the cellular chain complex of $X$. Since $H_n(X^n,X^{n-1})$ is free with basis in one-to-one correspondence with the $n$-cells of $X$, one can think of elements of $H_n(X^n,X^{n-1})$ as linear combinations of $n$-cells of $X$.

 The homology groups of this cellular chain complex are called the cellular homology groups of $X$, temporarily denoted by $H_n^{\text{CW}}(X)$.

Theorem 2.58. $H_n^{\text{CW}}(X)\cong H_n(X)$.

Proposition 2.59. Here are a few immediate applications:

(i)	$H_n(X)=0$ if $X$ is a CW complex with no $n$-cells.
(ii)	More generally, if $X$ is a CW complex with $k$ $n$-cells, then $H_n(X)$ is generated by at most $k$ elements.
(iii)	If $X$ is a CW complex having no two of its cells in adjacent dimensions, then $H_n(X)$ is free abelian with basis in one-to-one correspondence with the $n$-cells of $X$.

Proof.

Proof.

(ii)	$H_n(X)\cong \mathrm{Ker}{d}_n/\mathrm{Im}{d}_{n+1}$ and $\mathrm{Ker}{d}_n\subset H_n(X^n,X^{n-1})={\mathbb{Z}}^k$.
(iii)	The maps $d_n$ are automatically zero.

$\square$

Example 2.60. $H_i(\mathbb{C}{P}^n)\cong \{\begin{array}{ll}\mathbb{Z},& \text{for }i=0,2,4,\cdots ,2n\\ 0,& \text{otherwise}\end{array}$

Example 2.61. $S^n\times S^n$ with $n>1$, using the product structure consisting of a $0$-cell, two $n$-cells, and a $2n$-cell.

Proposition 2.62. Cellular Boundary Formula

$$d_n(e_{\alpha }^n)=\sum _{\beta }{d}_{\alpha \beta }{e}_{\beta }^{n-1}$$where $d_{\alpha \beta }$ is the degree of the map $S_{\alpha }^{n-1}\to X^{n-1}\to S_{\beta }^{n-1}$ that is the composition of the attaching map of $e_{\alpha }^n$ with the quotient map collapsing $X^{n-1}-e_{\beta }^{n-1}$ to a point.

Remark 2.63. In case $X$ is connected and has only one $0$-cell, then $d_1$ must be $0$, otherwise $H_0(X)$ would not be $\mathbb{Z}$.

Example 2.64. $M_g$: one $0$-cell, $2g$ $1$-cells, and one $2$-cell attached by the product of commutators $[a_1,b_1]\cdots [a_g,b_g]$. The associated cellular chain complex is $$0\to \mathbb{Z}\stackrel{d_2}{\to }{\mathbb{Z}}^{2g}\stackrel{d_1}{\to }\mathbb{Z}\to 0.$$Note that both $d_1$ and $d_2$ are $0$, then one can get $$H_n(M_g)=\{\begin{array}{ll}\mathbb{Z},& n=0,2\\ {\mathbb{Z}}^{2g},& n=1\end{array}.$$

Example 2.65. $N_g$: one $0$-cell, $g$ $1$-cells, and one $2$-cell attached by the word $a_1^2\cdots a_g^2$. The associated cellular chain complex is $$0\to \mathbb{Z}\stackrel{d_2}{\to }{\mathbb{Z}}^g\stackrel{d_1}{\to }\mathbb{Z}\to 0.$$Note that $d_1=0$ and $d_2(1)=(2,2,\cdots ,2)$, then one can get $$H_n(N_g)=\{\begin{array}{ll}\mathbb{Z},& n=0\\ {\mathbb{Z}}^{g-1}\oplus {\mathbb{Z}}_2,& n=1\\ 0,& n=2\end{array}.$$

Remark 2.66. These two examples illustrate the general fact that the orientablility of a close connected manifold $M$ of dimension $n$ is detected by $H_n(M)$, which is $\mathbb{Z}$ if $M$ is oriented and $0$ otherwise. This will be shown in the next chapter.

Definition 2.67. If $X$ satisfies ${\tilde{H}}_i(X)=0$ for all $i$, we call $X$ an acyclic space.

Example 2.68. The examples of $T^3=S^1\times S^1\times S^1$ and $K\times S^1$ are also amazing. One can refer to Hatcher P142.

Definition 2.69. Moore Spaces

 Given an abelian group $G$ and an integer $n\ge 1$, we can construct a CW complex $X$ such that $H_n(X)\cong G$ and ${\tilde{H}}_i(X)=0$ for $i\ne n$, commonly written $M(G,n)$ to indicate the dependence of $G$ and $n$.

Example 2.70. Real Projective Space $\mathbb{R}{\mathbf{P}}^{\mathbf{n}}$

 $d_k=\mathrm{deg}\mathrm{id}+\mathrm{deg}(-{\mathrm{id}}_{S^{k-1}})=1+(-1{)}^k$, thus the cellular chain complex for $\mathbb{R}{P}^n$ is $$0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\stackrel{0}{\to }\cdots \stackrel{2}{\to }\mathbb{Z}\stackrel{0}{\to }\mathbb{Z}\to 0,\text{if}n\text{is even}$$$$0\to \mathbb{Z}\stackrel{0}{\to }\mathbb{Z}\stackrel{2}{\to }\cdots \stackrel{2}{\to }\mathbb{Z}\stackrel{0}{\to }\mathbb{Z}\to 0,\text{if}n\text{is odd}$$From this it follows that $$H_k(\mathbb{R}{P}^n)=\{\begin{array}{ll}\mathbb{Z},& \text{for}k=0\text{and for}k=n\text{odd}\\ {\mathbb{Z}}_2,& \text{for}k\text{odd},0<k<n\\ 0,& \text{otherwise}\end{array}.$$

Euler Characteristic

Definition 2.71. For a finite CW complex $X$, the Euler characteristic $\chi (X)$ is defined to be the alternating sum ${\sum }_n(-1{)}^n{c}_n$ where $c_n$ is the number of $n$-cells of $X$.

Remark 2.72. The following result shows that $\chi (X)$ can be defined purely in terms of homology, and hence depends only on the homotopy type of $X$. In particular, $\chi (X)$ is independent of the choice of CW structure on $X$.

Theorem 2.73. $\chi (X)={\sum }_n(-1{)}^n\mathrm{rank}{H}_n(X)$.

Definition 2.74. Here the rank of a finitely generated abelian group is the number of $\mathbb{Z}$ summands when the group is expressed as a direct sum of cyclic groups.

 If $0\to A\to B\to C\to 0$ is a short exact sequence of finitely generated abelian groups, then $\mathrm{rank}B=\mathrm{rank}A+\mathrm{rank}C$.

Example 2.75. $\chi (M_g)=2-2g$ and $\chi (N_g)=2-g$.

Split Exact Sequences

Lemma 2.76. Splitting Lemma

  For a short exact sequence $0\to A\stackrel{i}{\to }B\stackrel{j}{\to }C\to 0$ of abelian groups the following statements are equivalent:

(a)	There is a homomorphism $p:B\to A$ such that $p\circ i=\mathrm{id}:A\to A$.
(b)	There is a homomorphism $s:C\to B$ such that $j\circ s=\mathrm{id}:C\to C$.
(c)	There is a an isomorphism $B\cong A\oplus C$ making a commutative diagram, where the maps in the lower row are the obvious ones, $a\mapsto (a,0)$ and $(a,c)\mapsto c$.

  If these conditions are satisfied, the exact sequence is said to be split.

Proposition 2.77. A retraction $r:X\to A$ gives a splitting $H_n(X)\cong H_n(A)\oplus H_n(X,A)$.

Example 2.78. The last proposition can be used to show the nonexistence of such a retraction in some cases.

(1)

For the Brouwer fixed point theorem, a retraction $D^n\to S^{n-1}$ would give $$H_{n-1}(D^n)\cong H_{n-1}(S^{n-1})\oplus H_{n-1}(D^n,S^{n-1}).$$

(2)

The mapping cylinder $M_f$ of a degree $m$ map $f:S^n\to S^n$ with $m>1$. If $M_f$ retracted onto the $S^n\subset M_f$ corresponding to the domain of $f$, we would have a split short exact sequence But this sequence does not split since $\mathbb{Z}$ is not isomorphic to $\mathbb{Z}\oplus {\mathbb{Z}}_m$ if $m>1$, so the retraction cannot exist.   The simplest case of degree $2$ map $S^1\to S^1$, $z\mapsto z^2$, this says that the Möbius band does not retract onto its boundary circle.

Mayer-Vietoris Sequences

Proposition 2.79. For a pair of subspaces $A,B\subset X$ such that $X$ is the union of the interiors of $A$ and $B$, this exact sequence has the form $$\cdots \to H_n(A\cap B)\stackrel{\Phi }{\to }{H}_n(A)\oplus H_n(B)\stackrel{\Psi }{\to }{H}_n(X)\stackrel{\partial }{\to }{H}_{n-1}(A\cap B)\to \cdots \to H_0(X)\to 0$$where $\Phi (x)=(x,-x),\Psi (x,y)=x+y$.

Proposition 2.80. There is also a formally identical Mayer-Vietoris sequence for reduced homology groups, obtained by augmenting the previous short exact sequence of chain complexes in the obvious way:

Remark 2.81. Mayer-Vietoris sequence can be viewed as analogs of the van Kampen theorem since if $A\cap B$ is path-connected, the $H_1$ terms of the reduced Mayer-Vietoris sequence yield an isomorphism $H_1(X)\cong \left(H_1(A)\oplus H_1(B)\right)/\mathrm{Im}\Phi$. This is exactly the abelianized statement of the van Kampen theorem, and $H_1$ is the abelianization of ${\pi }_1$ for path-connected spaces.

Remark 2.82. There are also Mayer-Vietoris sequences for decompositions $X=A\cup B$ such that $A$ and $B$ are deformation retracts of neighborhoods $U$ and $V$ with $U\cap V$ deformation retracting onto $A\cap B$.

Example 2.83. Take $X=S^n$ with $A$ and $B$ the northern and southern hemispheres, so that $A\cap B=S^{n-1}$. Then in the reduced Mayer-Vietoris sequence the terms ${\tilde{H}}_i(A)\oplus {\tilde{H}}_i(B)$ are zero, so we obtain isomorphisms ${\tilde{H}}_i(S^n)\cong {\tilde{H}}_{i-1}(S^{n-1})$.

Example 2.84. We can decompose the Klein bottle $K$ as the union of two Möbius bands $A$ and $B$ glued together by a homeomorphism between their boundary circles.

 Then $A,B$ and $A\cap B$ are homotopy equivalent to circles, so the interesting part of the reduced Mayer-Vietoris sequence for the decomposition $K=A\cup B$ is the segment $$0\to H_2(K)\to H_1(A\cap B)\stackrel{\Phi }{\to }{H}_1(A)\oplus H_1(B)\to H_1(K)\to 0.$$The map $\Phi$ is $\mathbb{Z}\to \mathbb{Z}\oplus \mathbb{Z}$, $1\mapsto (2,-2)$, since the boundary circle of a Möbius band wraps twice around the core circle. Since $\Phi$ is injective we obtain $H_2(K)=0$. Furthermore, we have $H_1(K)\cong \mathbb{Z}\oplus {\mathbb{Z}}_2$ since $\mathbb{Z}\oplus \mathbb{Z}=⟨(1,0),(1,-1)⟩$.

Homology with coefficients

Definition 2.85. The generalization consists of using chains of the form ${\sum }_i{n}_i{\sigma }_i$ where each ${\sigma }_i$ is a singular $n$-simplex in $X$ as before, but now the coefficients $n_i$ are taken to be in a fixed abelian group $G$ rather than $\mathbb{Z}$. Such $n$-chains form an abelian group $C_n(X;G)$ and there is the expected relative version $C_n(X,A;G)=C_n(X;G)/C_n(A;G)$.

  $\partial$ defined similarly satisfies ${\partial }^2=0$ and hence $C_n(X;G)$ and $C_n(X,A;G)$ form chain complexes. The resulting homology groups $H_n(X;G)$ and $H_n(X,A;G)$ are called homology groups with coefficients in $\mathbf{G}$.

  Reduced groups ${\tilde{H}}_n(X;G)$ are defined via the augmented chain complex $\cdots \to C_0(X;G)\stackrel{\epsilon }{\to }G\to 0$ with $\epsilon$ again defined by summing coefficients.

Proposition 2.86. All the theory we developed for $\mathbb{Z}$ coefficients carries over directly to general coefficient group $G$.

Lemma 2.87. If $f:S^k\to S^k$ has degree $m$, then $f_{\ast }:H_k(S^k;G)\to H_k(S^k;G)$ is multiplication by $m$.

  Then one could define $d_{\alpha \beta }$ in the cellular homology.

Proof.

Proof.

$\square$

Remark 2.88. The case $G={\mathbb{Z}}_2$ is particularly simple since one is just considering sums of singular simplices with coefficients $0$ or $1$, so by discarding terms with coefficient $0$ one can think of chains as just finite ‘unions’ of singular simplices.

  The boundary formulas also simplify since one on longer has to worry about signs. Since signs are an algebraic representation of orientation considerations, one can also ignore orientations. This means that homology with ${\mathbb{Z}}_2$ coefficients is often the most natural tool in the absence of orientability.

Example 2.89. It is instructive to see what happens to the homology of $\mathbb{R}{P}^n$ when the coefficient group $G$ is chosen to be a field $F$. The cellular chain complex is $$\cdots \stackrel{0}{\to }F\stackrel{2}{\to }F\stackrel{0}{\to }F\stackrel{2}{\to }F\stackrel{0}{\to }F\to 0.$$Hence if $F$ has characteristic $2$, for example $F={\mathbb{Z}}_2$, then $H_k(\mathbb{R}{P}^n;F)\cong F$ for $0\le k\le n$, a more uniform answer than with $\mathbb{Z}$ coefficients.

  On the other hand, if $F$ has characteristic different from $2$, then the boundary maps $F\stackrel{2}{\to }F$ are isomorphisms, hence $H_k(\mathbb{R}{P}^n;F)$ is $F$ for $k=0$ and for $k=n$ odd, and is zero otherwise.

Remark 2.90. In spite of the fact that homology with $\mathbb{Z}$ coefficients determines homology with other coefficient groups, there are many situations where homology with a suitably chosen coefficient group can provide more information than homology with $\mathbb{Z}$ coefficients.

Remark 2.91. In the next chapter we will see that there is a general algebraic formula expressing homology with arbitrary coefficients in terms of homology with $\mathbb{Z}$ coefficients.

Homology and Fundamental Group

Theorem 2.92. By regarding loops as singular $1$-cycles, we obtain a homomorphism $h:{\pi }_1(X,x_0)\to H_1(X)$. If $X$ is path-connected, then $h$ is surjective and has kernel the commutator subgroup of ${\pi }_1(X)$, so $h$ induces an isomorphism from the abelianization of ${\pi }_1(X)$ onto $H_1(X)$.

Classical Applications

Proposition 2.93. For an embedding $h:D^k\to S^n$, ${\tilde{H}}_i(S^n-h(D^k))=0$ for all $i$.

Proposition 2.94. For an embedding $h:S^k\to S^n$ with $k<n$, ${\tilde{H}}_i(S^n-h(S^k))$ is $\mathbb{Z}$ for $i=n-k-1$ and $0$ otherwise.

Proof.

Proof. Induction by $k$. Write $S^k$ as the union of hemispheres $D_{+}^k$ and $D_{-}^k$ intersecting in $S^{k-1}$. The Mayer-Vietoris sequence for $A=S^n-h(D_{+}^k)$ and $B=S^n-h(D_{-}^k)$, both of which have trivial reduced homology by the last proposition, then gives isomorphisms ${\tilde{H}}_i(S^n-h(S^k))\cong {\tilde{H}}_{i+1}(S^n-h(S^{k-1}))$.

$\square$

Corollary 2.95. Jordan curve theorem

  A subspace of $S^n$ homeomorphic to $S^{n-1}$ separates it into two components, and these components have the same homology groups as a point.

Corollary 2.96. An embedding $h:S^n\to S^n$ must be surjective.

Proof.

Proof. As in the last proof, consider the end of the Mayer-Vietoris sequence $${\tilde{H}}_0(A)\oplus {\tilde{H}}_0(B)\to {\tilde{H}}_0(S^n-h(S^{n-1}))\to {\tilde{H}}_{-1}(S^n-h(S^n)).$$For ${\tilde{H}}_0(S^n-h(S^{n-1}))\cong \mathbb{Z}$, we have $S^n-h(S^n)$ must be empty.

$\square$

Corollary 2.97. In particular, $S^n$ cannot be embedded in ${\mathbb{R}}^n$ since this would yield a nonsurjective embedding in $S^n$.

 A consequence is that there is no embedding ${\mathbb{R}}^m\hookrightarrow {\mathbb{R}}^n$ for $m>n$ since this would restrict to an embedding of $S^n\subset {\mathbb{R}}^m$ into ${\mathbb{R}}^n$.

 More generally there is no continuous injection ${\mathbb{R}}^m\to {\mathbb{R}}^n$ for $m>n$ since this too would give an embedding $S^n\hookrightarrow {\mathbb{R}}^n$.

Theorem 2.98. Invariance of Domain

 If $U$ is an open set in ${\mathbb{R}}^n$ and $h:U\to {\mathbb{R}}^n$ is an embedding, or more generally just a continuous injection, then the image $h(U)$ is an open set in ${\mathbb{R}}^n$.

Corollary 2.99. If $M$ is a compact $n$-manifold and $N$ is a connected $n$-manifold, then an embedding $h:M\to N$ must be surjective, hence a homeomorphism.

Proposition 2.100. An odd map $f:S^n\to S^n$, satisfying $f(-x)=-f(x)$ for all $x$, must have odd degree.

Proposition 2.101. Given a two-sheeted covering space $p:\tilde{X}\to X$, $$\cdots \to H_n(X;{\mathbb{Z}}_2)\stackrel{{\tau }_{\ast }}{\to }{H}_n(\tilde{X};{\mathbb{Z}}_2)\stackrel{p_{\ast }}{\to }{H}_n(X;{\mathbb{Z}}_2)\to H_{n-1}(X;{\mathbb{Z}}_2)\to \cdots$$ This is the long exact sequence of homology groups associated to a short exact sequence of chain complexes consisting of short exact sequences of chain groups $$0\to C_n(X;{\mathbb{Z}}_2)\stackrel{\tau }{\to }{C}_n(\tilde{X};{\mathbb{Z}}_2)\stackrel{p_{\#}}{\to }{C}_n(X;{\mathbb{Z}}_2)\to 0$$The map $p_{\#}$ is surjective since singular simplices $\sigma :{\Delta }^n\to X$ always lift to $\tilde{X}$, as ${\Delta }^n$ is simply-connected. Each $\sigma$ has in fact precisely two lifts ${\tilde{\sigma }}_1$ and ${\tilde{\sigma }}_2$. Because we are using ${\mathbb{Z}}_2$ coefficients, the kernel of $p_{\#}$ is generated by the sums ${\tilde{\sigma }}_1+{\tilde{\sigma }}_2$. So if we define $\tau$ to send each $\sigma$ to the sum of its two lifts, then the image of $\tau$ is the kernel of $p_{\#}$. Obviously $\tau$ is injective, so we have the short exact sequence indicated.

 Since $\tau$ and $p_{\#}$ commute with boundary maps, we have a short exact sequence of chain complexes, yielding the long exact sequence of homology groups.

 The map ${\tau }_{\ast }$ is a special case of more general transfer homomorphisms considered in the next chapter, so we will refer to the long exact sequence involving the maps ${\tau }_{\ast }$ as the transfer sequence.

Corollary 2.102. Borsuk-Ulam Theorem

 For every map $g:S^n\to {\mathbb{R}}^n$ there exists a point $x\in S^n$ with $g(x)=g(-x)$.

3Cohomology

Definition 3.1. Consider a chain complex $C$ of free abelian groups $$\cdots \to C_{n+1}\stackrel{\partial }{\to }{C}_n\stackrel{\partial }{\to }{C}_{n-1}\to \cdots$$The dual of $C_n$ is the cochain group $C_n^{\ast }=\mathrm{Hom}(C_n,G)$, and the dual of $\partial$ is the coboundary map $\delta :C_{n-1}^{\ast }\to C_n^{\ast }$. $$\cdots \leftarrow C_{n+1}^{\ast }\stackrel{\delta }{\leftarrow }{C}_n^{\ast }\stackrel{\delta }{\leftarrow }{C}_{n-1}^{\ast }\leftarrow \cdots$$It follows from $\partial \partial =0$ that $\delta \delta =0$, and we can define the cohomology group $H^n(C;G)=\mathrm{Ker}\delta /\mathrm{Im}\delta$.

Definition 3.2. A free resolution of an abelian group $H$ is an exact sequence of free groups $$\cdots \to F_1\stackrel{f_1}{\to }{F}_0\stackrel{f_0}{\to }H\to 0$$Appply $\mathrm{Hom}(-,G)$ and we get a chain complex $$\cdots \leftarrow F_1^{\ast }\stackrel{f_1^{\ast }}{\leftarrow }{F}_0^{\ast }\stackrel{f_0^{\ast }}{\leftarrow }{H}^{\ast }\leftarrow 0$$We may lose exactness, and use the temporary notation $H^n(F;G)=\mathrm{Ker}{f}_{n+1}^{\ast }/\mathrm{Im}{f}_n^{\ast }$.

Lemma 3.3. For free resolutions $F$ and $F^{\prime }$ of $H$, there are canonical isomorphisms $H^n(F;G)\cong H^n(F^{\prime };G)$ for all $n$.

Every $H$ has a free resolution of the form $0\to F_1\to F_0\to H\to 0$. Suppose $H$ is generated by $\{h_i\}$, and let $F_0$ be the free abelian group with basis $\{h_i\}$ and $f_0:F_0\to H,h_i\mapsto h_i$. Let $F_1=\mathrm{Ker}{f}_0$, which is a subgroup of a free abelian group thus also free, and $f_1$ be the inclusion $F_1\hookrightarrow F_0$. For this resolution we get $H^n(F;G)=0$ for $n>1$, which is hence also true for all free resolutions. Thus only $H^1(F;G)$ is nontrivial, written as $\mathrm{Ext}(H,G)$.

Theorem 3.4. Universal Coefficient Theorem for Cohomology There is a split exact sequence $$0\to \mathrm{Ext}(H_{n-1}(C),G)\to H^n(C;G)\stackrel{h}{\to }\mathrm{Hom}(H_n(C),G)\to 0$$

Proposition 3.5.

•	$\mathrm{Ext}(H\oplus H^{\prime },G)=\mathrm{Ext}(H,G)\oplus \mathrm{Ext}(H^{\prime },G)$.
•	$\mathrm{Ext}(H,G)=0$ if $H$ is free.
•	$\mathrm{Ext}({\mathbb{Z}}_n,G)=G/nG$.

Corollary 3.6. If $H_n$ and $H_{n-1}$ of a chain complex $C$ of free abelian groups are finitely generated, with torsion subgroups $T_n$ and $T_{n-1}$, then $H^n(C;\mathbb{Z})\cong (H_n/T_n)\oplus T_{n-1}$.

Lemma 3.7. If a sequence of abelian groups $A\stackrel{i}{\to }B\stackrel{j}{\to }C\to 0$ is exact, then so is $A\otimes 1\stackrel{i\otimes 1}{\to }B\otimes 1\stackrel{j\otimes 1}{\to }C\otimes 1\to 0$.

Lemma 3.8. For free resolutions $F$ and $F^{\prime }$ of $H$, there are canonical isomorphisms $H_n(F\otimes G)\cong H_n(F^{\prime }\otimes G)$ for all $n$.

The group $H_n(F\otimes G)$, which depends only on $H$ and $G$, is denoted ${\mathrm{Tor}}_n(H,G)$. Since a free resolution $0\to F_1\to F_0\to H\to 0$ always exists, only ${\mathrm{Tor}}_1(H,G)$ is nontrivial, and is written as $\mathrm{Tor}(H,G)$.

Theorem 3.9. Universal Coefficient Theorem for Homology There is a natural exact sequence that splits though not naturally $$0\to H_n(C)\otimes G\to H_n(C;G)\to \mathrm{Tor}(H_{n-1}(C),G)\to 0$$

Proposition 3.10.

•	$\mathrm{Tor}(A,B)=\mathrm{Tor}(B,A)$.
•	$\mathrm{Tor}({⨁}_i{A}_i,B)={⨁}_i\mathrm{Tor}(A_i,B)$.
•	$\mathrm{Tor}(A,B)=0$ if $A$ or $B$ is free, or more generally torsionfree.
•	$\mathrm{Tor}(A,B)=\mathrm{Tor}(T(A),B)$ where $T(A)$ is the torsion subgroup of $A$.
•	$\mathrm{Tor}({\mathbb{Z}}_n,A)=\mathrm{Ker}(A\stackrel{n}{\to }A)$.

Corollary 3.11. For finitely generated $A$ and $B$, $\mathrm{Tor}(A,B)\cong T(A)\otimes T(B)$.

Corollary 3.12. (a) $H_n(X;\mathbb{Q})\cong H_n(X)\otimes \mathbb{Q}$, so when $\mathrm{rank}{H}_n(X)<\infty$, it equals ${\dim }_{\mathbb{Q}}{H}_n(X;\mathbb{Q})$.
(b) If $H_n(X)$ and $H_{n-1}(X)$ are finitely gererated, then for $p$ prime, $H_n(X;{\mathbb{Z}}_p)$ consists a ${\mathbb{Z}}_p$ summand for: each $\mathbb{Z}$ summand of $H_n(X)$, each ${\mathbb{Z}}_{p^k}$ summand of $H_n(X)$, each ${\mathbb{Z}}_{p^k}$ summand of $H_{n-1}(X)$.

Remark 3.13. Here is a similar result given in Exercise 2.2.43:
If $X$ is a CW complex with finitely many cells in each dimension, then $H_n(X;G)$ is the direct sum of: a $G$ for each $\mathbb{Z}$ summand of $H_n(X)$, a $G/mG$ for each ${\mathbb{Z}}_m$ summand of $H_n(X)$, a $\mathrm{Ker}(G\stackrel{m}{\to }G)$ for each ${\mathbb{Z}}_m$ summand of $H_{n-1}(X)$.

Definition 3.14. A local orientation of $M$ at $x$ is a choice of generator ${\mu }_x$ of $H_n(M|x)\cong \mathbb{Z}$, and an orientation of $M$ is a function $x\mapsto {\mu }_x$ assigning to each $x\in M$ a local orientation ${\mu }_x\in H_n(M|x)$ satisfying: each $x\in M$ has a neighborhood $B$, such that all ${\mu }_y$ at $y\in B$ are the images of one generator ${\mu }_B\in H_n(M|B)\cong H_n({\mathbb{R}}^n|B)$ under the natural maps $H_n(M|B)\to H_n(M|y)$. $M$ is orientable if an orientation exists.