## HATCHER ALGEBRAIC TOPOLOGY HOMEWORK SOLUTIONS

• June 24, 2019

The fundamental group and the first homology group. Through this quarter we will construct, prove properties about, and study applications of three such invariants:. I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty. You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions. Statement of the cohomology ring of projective spaces, an application to the Borsuk-Ulam theorem.

If you tend to scratch out or erase incorrect parts of solutions, do a rough draft or type your homework. Ragged margins, multi-column and otherwise over-dense formats, and unstapled sheets are unacceptable. It is due on Thursday, November 1, at 5: Cheating on exams is unacceptable. The homology of real projective space.

Degrees of maps between spheres.

Your grade will be determined by your performance on homework assignments and your overall classroom engagement: Ralph Cohen Course Assistant. Additional notes completing algebraiv construction of the Barycentric subdivision proposition.

Use punctuation and conjunctions to indicate your flow of thought rather than arrows or telepathy. Religious holidays are not an excuse for late homework!

Any cheating during midterms or finals will result in you failing the exam. Homework must homfwork neat, well-organized, and legible. Shoot for lucidity rather than terseness.

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Co-chain complexes and cohomology. Here is the midterm exam.

Allen Hatcher’s Algebraic Topologyavailable for free download here. Proof of excision via barycentric subdivision, the equivalence of singular and simplicial homology and the 5-lemma. Cup product in the relative setting. The homology of real projective space.

That said, the work you hand in must be your own. In addition to formal prerequisites, we will use a number of notions and concepts without much explanation.

Graphs and sub-groups of free groups. A basic class of topological spaces we will apply our discussion to is the class of CW complexes.

Calculation of the cup product on projective spaces, via a reduction to Euclidean spaces. The cellular boundary formula, and applications to real projective space.

# Math A: Algebraic Topology

I will drop your lowest homework score to allow for missed assignments or for assignments that pose special difficulty. Homology of good pairs. Students who may need accommodations because of a documented disability should meet with me privately within the first week of classes.

Statement of the cohomology ring honework projective spaces, an application to the Borsuk-Ulam theorem.

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# Math Topology I | David Shea Vela-Vick

Through this quarter we will construct, prove properties about, and study applications of three such invariants:. Please submit your completed exam as a pdf file via email to both Prof. The correspondence theorem, Deck transformations, solutiojs. The best way to avoid any suspicion of copying is to write out your homework solutions on your own, not whilst working with a friend. It is due on Thursday, Dec. More on the degree of a map. Classical applications Brouwer’s fixed point theorem, Jordan-Brouwer curve separation.

Here is the Final Exam.

## Math 7510: Topology I

You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions. Our course will primarily use Chapters 0, 1, 2, and 3. Ragged margins, multi-column and otherwise over-dense formats, and unstapled sheets are unacceptable.