I.2 Categories 29 3. For every pair of morphisms ( f ,g) with f ∈ C ′ (X ′ ,Y ′ ) and g ∈ C ′ (Y ′ ,Z ′ ), the morphism obta<strong>in</strong>ed through composition <strong>in</strong> C ′ , namely, gf ∈ C ′ (X ′ ,Z ′ ), co<strong>in</strong>cides with the morphism gf ∈ C(X ′ ,Z ′ ) obta<strong>in</strong>ed through composition <strong>in</strong> C. A subcategory C ′ of C is called full if, for every pair of objects (X ′ ,Y ′ ) of C ′ , the sets C ′ (X ′ ,Y ′ ) and C(X ′ ,Y ′ ) co<strong>in</strong>cide. The category CTop is a subcategory of Top × Top. The set Top(X,Y ) of all maps from X to Y has a very important relation called homotopy. LetI be the closed <strong>in</strong>terval [0,1]. Twomapsf ,g: X → Y are homotopic when there is a map H : X × I → Y such that H(−,0)= f and H(−,1)=g. Iff is homotopic to g, we write f ∼ g. For <strong>in</strong>stance, the maps f ,g: I → I given by f = 1I (that is to say, ∀x, f (x)=x) and the constant map g from I to the po<strong>in</strong>t 0 ∈ I (<strong>in</strong> other words, ∀x, f (x) =0) are homotopic; <strong>in</strong> fact, by construct<strong>in</strong>g the map H : I × I → I , (∀s,t ∈ I) H(s,t)=(1 − t)s, it is clear that H(−,0)= f and H(−,1)=g. A homotopy may be <strong>in</strong>dicated also by the notation ft : X → Y ,wheret is the parameter t ∈ I and, if H(x,t) is the homotopy function, then ft(x) := H(x,t). Consequently, f0 ∼ f1. By Theorem (I.1.37), a homotopy H : X × I → Y from f to g determ<strong>in</strong>es a map ˆH : I → M(X,Y ), thatistosay,apath <strong>in</strong> M(X,Y ) that l<strong>in</strong>ks f to g. Conversely,if X is compact Hausdorff, a path ˆH : I → M(X,Y ) determ<strong>in</strong>es a map H : X × I → Y , with H(−,0)= f and H(−,1)=g (Theorem (I.1.39)), <strong>in</strong> other words, a homotopy from f to g. (I.2.1) Lemma. The homotopy relation is an equivalence relation. Proof. Clearly, any map f is homotopic to itself. Suppose now that f ∼ g; this means that there is a map consider the map H : X × I → Y , H(−,0)= f , H(−,1)=g; H ′ : X × I → Y , (∀x ∈ X)(∀t ∈ I) H ′ (x,t)=H(x,1 − t). It is immediate to show that H ′ (−,0)=g and H ′ (−,1)= f and so, g ∼ f . We f<strong>in</strong>ally prove that, if f ∼ g and g ∼ h,then f ∼ h. Consider the functions H : X × I → Y , H(−,0)= f , H(−,1)=g, G: X × I → Y , G(−,0)=g , G(−,1)=h. Let us def<strong>in</strong>e a map K : X × I → Y by the conditions � 1 H(x,2t) , 0 ≤ t ≤ (∀x ∈ x)(∀t ∈ I) K(x,t)= 2 G(x,2t − 1) , 1 2 ≤ t ≤ 1. This map K has the required properties. �
30 I Fundamental Concepts The quotient set obta<strong>in</strong>ed from Top(X,Y ) and the relation ∼ is denoted by [X,Y ]; its elements are equivalence classes for homotopy of maps, or homotopy classes of maps from X to Y. The category HTop,thehomotopy category associated with Top, has topological spaces for objects and its morphisms are homotopy classes of maps: the set of morphisms is, therefore, HTop(X,Y)=[X,Y]. When deal<strong>in</strong>g with based maps f ,g ∈ Top ∗((X,x0),(Y,y0)), we say that f ∼ g if there is a map H : X × I → Y such that (∀t ∈ I) H(x0,t)=y0 , H(−,0)= f , and H(−,1)=g. In this case, we use the notation [X,Y]∗ = Top ∗(X,Y)/∼. Relative homotopy <strong>in</strong> CTop is a useful concept: two maps of pairs f ,g: (X,A) → (Y,B) are homotopic relative to X ′ ⊂ X if f |X ′ = g|X ′ and there exists a map such that H : (X × I,A × I) −→ (Y,B) (∀x ∈ X) H(x,0)= f (x) H(x,1)=g(x) (∀x ∈ X ′ , ∀t ∈ I) H(x,t)= f (x)=g(x). This be<strong>in</strong>g the case, we denote the relative homotopy from f to g with f ∼ relX ′ g and we say that f is homotopic to g rel X ′ . If X ′ = /0, we have a free homotopy <strong>in</strong> CTop. In the category Top ∗ the based homotopy co<strong>in</strong>cides with the homotopy relative to the base po<strong>in</strong>t. We say that two spaces X,Y ∈ Top are of the same homotopy type (or simply, type) iftherearemapsf : X → Y and g: Y → X such that gf ∼ 1X and fg∼ 1Y ; the map f is called a homotopy equivalence. The “functions” between categories are called functors, which take objects to objects and morphisms to morphisms. Specifically, a covariant functor or simply functor F : C → C ′ is a relation between these two categories such that 1. (∀X ∈ C) F(X) ∈ C ′ 2. (∀ f ∈ C(A,B)) F( f ) ∈ C ′ (F(A),F(B)) 3. (∀ f ∈ C(A,B))(∀g ∈ C(B,C)) F(gf)=F(g)F( f ) 4. (∀A ∈ C) F(1A)=1FA
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88 II Simplicial Complexes Exercise
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Chapter III Homology of Polyhedra I
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III.1 The Category of Polyhedra 101
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III.2 Homology of Polyhedra 109 Now
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152 IV Cohomology (IV.1.1) Theorem.
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154 IV Cohomology (IV.1.2) Remark.
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156 IV Cohomology If, starting from
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158 IV Cohomology When we apply thi
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160 IV Cohomology groups Q and R ar
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164 IV Cohomology Since g and π r
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166 IV Cohomology It is easily prov
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Chapter V Triangulable Manifolds V.
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210 VI Homotopy Groups Exercises 1.
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212 VI Homotopy Groups Proof. Since
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214 VI Homotopy Groups is precisely
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216 VI Homotopy Groups The group π
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References 1. J.W. Alexander - A pr
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Index H-space, 219 n-simple, 225 ab
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Index 243 Euclidean, 43 join, 47 su