Message-ID: <902850120.3839.1566178083543.JavaMail.tomcat@wikis-live> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_3838_2141551110.1566178083542" ------=_Part_3838_2141551110.1566178083542 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Topologie II WiSe 14/15

# Topologie II WiSe 14/15

Winter Term 2014/2015

Welcome to the course website! This course on algebraic topology is taug= ht by Pavle Blagojevi=C4=87  and Holger Reich and is a continuation of Topologie I. Although it i= s helpful to have taken Topologie I, it is not necessary.

Second Written Exam

Here are the results= of the second written exam. If there are questions, please come = by Albert's office on Tuesday at 4PM.

# Course Description

We understand this course as a comprehensive beginners course in algebra= ic topology. Although it is helpful to have taken Topologie I to foll= ow the present course, it is not necessary. The aim of Topologie II = is for you to thoroughly understand and be able to apply all n= otions of homology and cohomology, the cup and cross product, as well as so= me results on duality (for instance Poincar=C3=A9 duality). The course will= roughly be structured as follows (as time permits):

• Categories and functors, chain complexes

• Singular homology, chain homotopy

• Mayer-Vietoris Sequence, Jordan Curve Theorem

• Reduced homology, relative homology, Alexander's theorem

• Simplicial homology

• Degrees, Euler characteristic, Lefschetz number, Lefschetz fixed poi= nt theorem

• CW complexes

• Cellular homology

• Eilenberg=E2=80=93Steenrod axioms

• K=C3=BCnneth Theorem

• Universal Coefficient Theorem

• Singular cohomology, simplicial cohomology

• Cup product

• Cross product, topological manifolds

• Poincar=C3=A9 Duality

• Alexander Duality

• Manifolds with boundary

We recommend the books by J. Munkres ("Elements of Algebraic Topology", = Addison-Wesley 1984) and A. Hatcher ("Algebraic Topology", Cam= bridge U Press 2002, also online) and the succinct lecture notes by J. P. M= ay ("A Concise Course in Algebraic Topology", online).

# Contact

 Contact Office Hours: Lecture Pavle Blagojevi=C4=87  and Holger Reich blagojevic(at)math.fu-berlin.de, holger.re= ich(at)fu-berlin.de TBA Tutorial Albert Haase a.haase(at)fu-berlin.de Mon 13-14, 002, Arnimallee 2

# Lectures

 Lectures Wed 10:15 - 11:45 HS 001, Ar= nimallee 3 12:15 - 13:45 SR 031, Arnimallee 6

# Tutorials and Problems=

 Tutorials Mon 14:15 - 15:45 SR 031, Arnimallee 6

In the tutorial, we will learn some things that will help us better unde= rstand the lecture, expand on topics from class, and occasionally review ex= ercises.

Every now and then a sheet will appear on this website containing exerci= ses, some of which we highly recommend. Do the other exercises if they seem= challenging enough or if you don't have an idea of how to solve them immed= iately. Solutions to select exercises will appear on this website. In order= to encourage you to solve exercises, we will (1) periodically ask students= to present solutions to an exercise in the tutorial and (2) base parts of = the exam on the exercises.

Course requirements are the following: (1) You must actively participate= in the course. (2) You must pass an exam at the end of the semester which = alone will determine your grade. Th= e first written exam will take place on March 3 from 2-4 PM in "Gro=C3=9Fer= H=C3=B6rsaal Informatikgeb=C3=A4ude Takustra=C3=9Fe 9". The second written exam will take place on Apr= il 13 from 10AM - 12PM in "Seminarraum Animallee 2".

## Exercise Sheets

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