I started by say something about the purpose of this course, and recall the definitions of the basic notions of
homotopy, homotopy equivalences, retractions and deformation retractions and mapping cylinders (Hatcher p.1-3). Then I defined pushouts
(following the short note by Boardman (there is link to this note under "detailed teaching plan"). Then I introduced CW complexes
following Hatcher 5-7. I also stated the Lemma on. page 74 in May, that the pushout formed by the inclusion of subcomplex and a cellular map defined on this subcomplex can be made into a CW complex. I then defined the homotopy extension property (HEP) (def. p. 14 in Hatcher) and I talked about cofibrations (following May) giving the definition on p.43 and explaining how we can construct a universal test diagram for cofibrations using mapping cylinders (p. 44 , section 2 of chapter 6, May) and what this means when we have a pair (X, A) with A a closed subset of X. I then proved proposition 0.16, 0.17 and 0.18 in Hatcher. I then jumped back to May giving the definition of NDR-pair (p. 45), proving that if f:X-> Y then the pair (M_f, X) is an NDR-pair, then I stated and explained (without giving a complete proof) the Theorem at the bottom of p. 45, and we thus concluded that (M_f, X) satisfied HEP. I then turned to Hatcher proving 0.19, 0.20 and 0.21.
Next I moved to section 4.1 p. 339 and I have covered more or less the stuff up to CW approximation on p. 352 (however I have only proved a (minor) part of Theorem 4.3 (exactness at one place) and I have also skipped the proof of Lemma 4.10. I then turned to section 4.2, p. 375 about fiber bundles where I gave the defintion and proved Theorem 4.41. I the jumped to May chapter 7, and covered the three first sections in this chapter. I skipped section 4, but also covered section 5 and 6. Next week I start talking about CW approximation (Hatcher p. 352). This is what I have done so far (16/2).
In the subsection about CW approximation I have covered everything apart from Proposition 4.21 (which we will leave out from the syllabus and Proposition 4.22 (which is part of the syllabus but you have to read this by your own). In the subsection about excision (4.2) the statement of Theorem 4.23 is part of the syllabus (but I did not lecture the proof so I leave the proof out from the syllabus).I also lectured the corollaries 4.24, 4.25 and Proposition 4.28 (with proofs) together with Example 4.29 (which are part of the syllabus). Then I jumped to section about fiber bundles ones again and I defined the notion of fiber bundle (p. 376-377 ) gave the example 4.44, stated Proposition 4.48, but I left the out (read it your self!). Finally I lectured Example 4.53 and 4.54.
Next I started on 1,2 Van Kampens Theorem . I talked about free products of groups formulated Van Kampens Theorem, proved that S^n is simply connected when n>1. Looked at the examples 1.21 and 1.22, and proved Van Kampens Theorem. I then jumped to the subsection about applications to cell complexes and lectured everything here apart from example 1.29. Then I jumped to 1.3 and started to lecture about covering spaces. Here I have lectured almost everything from p. 56 to the middle of p. 68 (up to "representing covering spaces.."). I did not lecture example 1.35 instead I explained why the construction on p. 59 gives a contractible (hence universal) covering space of wedge of two circles. Then I jumped to p. 70 (deck transformation and group actions) and I lectured from p. 70 to p. 75 (including example 1.43).
On Wednesday March. 23. I started on section 2.1 defining sigma-complexes. I will proceed with this section Monday 28. defining simplicial and singular homology.
Jeg har n? gjennomg?tt om simplisial og singul?r homologi teori fra og med side 102 eksempel 2.36 p? side 141. Jeg har videre gjennomg?tt eksempel 2.42 (side 144). Jeg har videre gjennomg?tt om Eulerkarakteristikk (fra midt p? side 146 til midt p? side 147). Onsdag 4/5 forklarte jeg ogs? hvordan vi lager Mayer Vitoris sekvensen i absolutt og redusert homologi. (Midt p? side 149 til midt p? side 150). Mandag 9/5 vil jeg bruke sekvensen til ? regne ut noen homologigrupper. Jeg hopper s? til side 160, men overlater til dere ? lese om aksiomer for homologi og om kategorier og funktorer.
Jeg vil s? g? igjennom stoffet fra 2A (sammen hengen mellom fundamental gruppe og 1. homologi gruppe) og 2B fra side 169 til midt p? side 170
(dropper eksempel 2B:2). Jeg gjennomg?r s? fra og med Thm. 2B:3 og frem til Borsuk-Ulam teoremet og stopper pensum her (midt p? side 174). Regner med ? ha kommet hit 11/5. Det blir s? utvalgte oppgaver uke 20, 21 og ev. mandag 30/5. Det blir ingen undervisning etter dette.