This publication is meant to function a textbook for a direction in algebraic topology before everything graduate point. the most issues lined are the type of compact 2-manifolds, the basic workforce, masking areas, singular homology conception, and singular cohomology concept. those subject matters are built systematically, keeping off all unecessary definitions, terminology, and technical equipment. at any place attainable, the geometric motivation in the back of a number of the strategies is emphasised. The textual content contains fabric from the 1st 5 chapters of the author's prior publication, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of- print SINGULAR HOMOLOGY thought (GTM 70). the cloth from the sooner books has been conscientiously revised, corrected, and taken modern.
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Extra resources for A basic course in algebraic topology
1) 0 ˇ ˇ i Di D : 0 i D1 Let f W „ ! B a projective family with dim B D 1, „ smooth and K „b ample for all b 2 B. Further let X D „b0 for some b0 2 B a singular ﬁber and let W ‡ ! „ P X D Xz C riD1 i Fi where be an embedded resolution of X Â „. Finally let Y D Xz is the strict transform of X and F i are exceptional divisors for . We are interested in ﬁnding conditions that ˇ are necessary for K Y to remain ample (cf. 6)). Let Ei WD Fi ˇXz be the exceptional divisors for W Xz ! X and for simplicity ˇof computation, assume that the E i are irreducible.
Show that if f W Y ! X / is a divisor on X and G is a divisor on Y , then Chapter 2. 1) If D is big (resp. 2) If D is big (resp. pseudo-effective) and F is effective and its support contains all f -exceptional divisors, then f 1 D C F is big (resp. 3) Show that if G G 0 , then f G f G 0 . Deduce from this that if G is big (resp. 4) Give an example where G is not big, but f G is big. 12. X / such that Di 0, D1 Q D2 and D1 ^ D2 D 0. D 3 / has a multiple which is mobile. 13. X /. Show that if k is minimal, then r 1 ; : : : ; rk are linearly independent over Q.
As this ˇ computation is local near P the only relevant issue about ˇ the ampleness of KXz C E ˇXz is whether it is ample in a neighbourhood of E X WD E ˇXz . K Xz C EX / EX is positive. 4. Let Z be a smooth projective surface with non-negative Kodaira dimension and Z an effective divisor. K Z C / C > 0 for every proper curve C Z, then KZ C is ample. Chapter 3. d 3/ as EX is isomorphic to a plane curve of degree d . Again, we obtain the same condition as above and thus conclude that K ‡0 is ample if and only if d > 3.
A basic course in algebraic topology by Massey