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These invariants turn out to be rather computable, and they allow for some immediate geometric applications. We will establish some key properties of these homology groups like the homotopy-invariance and the excision theorem. A convenient variant is provided by singular homology with coefficients - a framework which makes necessary a short discussion of basic homological algebra including tensor and torsion products.

## Preparatory videos for Applied Algebraic Topology

For CW complexes, there is also the more combinatorial cellular homology theory. The course culminates in a proof that singular homology and cellular homology agree on CW complexes. This allows for more explicit calculations in examples of interest e. In the sequel Algebraic Topology II we discuss a bit more advanced subjects like cohomology theory.

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We will also discuss some basic concepts from homotopy theory. The exam is a written exam on all material treated in the course.

- Course on Algebraic Topology I (first semester 2013/2014)!
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You are not allowed to use notes. The regular homework assignments can either be handed in by email or in person before class. Homework is optional can only increase your final grade. The retake exam is on Wed Feb 27, in Ruppert It is optional and may be used to replace your exam grade in the above formula, if your home institution agrees.

Prerequisites - Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, along the lines of these notes by A. Hatcher and maybe a first encounter with the fundamental group.

For those who haven't seen this before, the " Intensive course on Categories and Modules" is recommended. Recordings of all the lectures will be available here , you'll need the password: Xz4F. Solutions to selected exercises Solutions to Selected exercises by Jorge.

## Lectures on Algebraic Topology

Solutions to Sagave's exercises week 4. Additional exercises Additional practice exercises by Jorge.

watch Additional practice exercises Lecture 9 by Jorge. The Hawaiian earring has a non-free fundamental group, see an old paper by Bart de Smit Leiden. Finite topological spaces are surprisingly rich and complicated. There are 9 non-equivalent topologies on the 3-element set.

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As the number of elements grows, so does the number of topologies. Roughly speaking they are Abelian groups measuring in how many distinct ways an n sphere fits into an m sphere. The answer is a baffling mix of order and chaos. Much remains to be done!