Date: Sat, 3 Jun 2023 11:35:28 +0300 (EEST) Message-ID: <252794517.22310.1685781328882@wiki-1.it.helsinki.fi> Subject: Exported From Confluence MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_Part_22309_1504720056.1685781328882" ------=_Part_22309_1504720056.1685781328882 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/exported.html Introduction to algebraic topology, fall 2013

# Introduction to algebraic topology, fall 201= 3

10 sp.

### Type

The course is intended to be the first introduction to the singular homo= logy theory and homological methods in algebraic topology.

The main idea of algebraic topology is to study topological problems usi= ng suitable algebraic invariants. These invariants en= able one to convert a given difficult topological problem into algebraic pr= oblem, which is easier to solve than the original problem. For instance con= sider a very natural question - is plane R^2 homeomorphic to a space R^3? M= ore generally can R^n be homeomorphic to R^m when n is not equal to m? Intu= itively it is clear that the answer should be 'no', but it is surprisingly = difficult to actually prove this precisely. Algebraic methods, such as homo= logy, turn out to be extremely efficient in the course of studying this and= other similar questions.

We start off with the brief excursion to the world of simplices, simplic= ial methods and Delta-complexes. Then we move on to the main subject of the= course - construction of the singular homology theory. We go through all t= he essential properties of the singular homology and apply them in order to= prove the classical topological results such as the Invariance of Domain, = Brouwer's fixed-point theorem, Brouwer-Jordan separation theorem, the main = theorem of Algrebra and others. We define and study the notion of the degre= e of the mapping. If schedule permits, we will also talk about CW-comp= lexes, cellular homology and the classification of compact 2-manifolds.&nbs= p;

For more detailed exposure, see Foreword. <= /p>

### Prerequis= ites

Linear algebra I, Algebra I, Topology I or corresponding courses. Topolo= gy II is useful, but not necessary. However, the notion of a general topolo= gical space and basic notions of topology -such as compactness, connectedne= ss, quotient space et cetera, should be familiar. We will briefly revisit t= he basic definitions and results during the lectures.

### Contents =

1. Simplices, simplicial complexes and Delta-complexes.
2. Homological algebra - abelian free groups, chain complexes, homology, l= ong exact homology sequence.
3. Singular homology theory - definition, basic properties, long exact hom= ology sequences of pairs and triples, homotopy axiom, excision, mayer-viato= ris sequence, computational devices, homology of the spheres, the degree of= the map.
4. Applications - Brouwer fixed-point theorem, Jordan-Brouwer separation t= heorem, Invariance of Domain, generalizations to the theory of manifolds, H= airy Ball Theorem, Fundamental Theory of Algebra.
If time permits:
5. CW - complexes and cellular homology. Classification of compact 2-manif= olds.

### Lecture= material

Foreword

Part II - Chain complexes and homology groups

Notes - Parts I-III last updated on 8.12.

Bonus (won't be asked in exam):

CW-complexes and cellular homology

### Summaries of the topics of the second part of the = course:

Homological algebra

Singular homology

Simplicial homology

List of applications

### Lectures

Weeks 36-42 and 44-50, Tuesday 12-14 in room B321 and Wednesday 12-14 in= room C122. Two hours of exercise classes per week.

### Exams

Two middle exams during fall.

Second exam of the course is December 13 at 12.30-16.30 in Exact= um, CK111. Contact the lecturer for any information and questions.
You = do not have to enroll.

First middle exam is TU 22.10 at 12-16 at A111 or B123, during official general exam of the department.

Enroll at the office latest 14.10! Contact the lecturer= if you cannot attend first middle exam.

Contents of the first exam - Part I and Algebra part of Part II - everyt= hing up to the page 122, simplicial chains is the first topic NOT in the ar= ea of the 1st exam.

Exercise wise - everything covered in exercises 1-6

It is also possible to complete the course via general exam in the end o= f the course.

Bonus points for the exercises: 25% - 2 point, 40% - 3 points, 50% - 4 p= oints, 60% - 5 points, 75% - 6 points.

### Bibliograp= hy

`Hatcher, A.: Algebraic Topology, available online at http://www.math.cornell.edu/~hatcher/AT/AT.pdf  `
```Maunder, C.R.F: Algebraic topology - Van Nostrand Reinhold Company, 19=
70.```
`Spanier, E.H: Algebraic Topology - McGraw-Hill, 1966.`
```Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology - Princ=
eton University Press, 1952.```

Short video on Klein's bottle - http://www.youtub= e.com/watch?v=3DE8rifKlq5hc

### Registration

Did you forget to register?  What to do?

### Exerci= se session:

Group Weekday Time Place Teacher
1.  Tuesday 14-16  B321  Aleksandr Pasharin

### Lecture log<= /h3> 3. 9 - Introductory speech " what is algebraic topology and why we need = it ". Recollection of the elementary linear algebra (Part I, section 1). De= finitions of affine and convex subsets.4.9 - Affine and convex sets. Affine and convex hull. Affinely independe= nt subsets, simplices. 10.9 - Standard simplices. Affine mappings. Boundary and interior points= of a simplex. Topology - basic definitions, continuous mappings, homeomorp= hisms. Subspaces. Product topology. Metric spaces.11.9 - Normed spaces. Standard topology on finite-dimensional vector spa= ces and simplices. Compactness and connectedness.17.9 - Closure, interior and boundary. Theorem 3.19. Simplicial complexe= s - definition and basic properties. Weak topology.18.9 - More on weak topology of simpicial complexes. Subdivisions. Baryc= entric subdivision.24.9 - Essential properties of the barycentric subdivision. Proposition = 4.21. Continuous mappings between polyhedra. Simplicial mappings. The conce= pt of homotopy and its basic properties.25.9 - Homotopy equivalences and classification of spaces up to a homoto= py type. Lemma 5.8 and Lemma 5.9. Simplicial approximation theorem and its = consequences.1.10 - Introduction to Delta-complexes. Quotient spaces and quotient map= pings. 2.10 - Delta-complexes - formalities, construction of the polyhedron, ex= amples. "Cut and glue" technique (example 6.18). Short video about Klein's = bottle. Algebra - concepts and examples of abelian groups and homomorphisms= of groups.8.10 - Quotient groups, factorization and isomorphism theorems for abeli= an groups. Linear combinations, linearly independent sets and basis in abel= ian groups. Finitely supported families. Construction of free abelian group= s with given basis. 9.10 - Direct product and direct sum, their essential properties. Simpli= cial chains and singular chains. 15.10 - Boundary operator. Motivation for it through the notion of orien= tation. The essential property of boundary operator - Theorem 9.11. Chain c= omplexes in general. Cycles, boundary and homology groups for chain complex= es. Simplicial and singular homology groups. 16.10 - Calculation of simplicial homologies - Examples 9.4. - 9.7.29.10 - Chain mappings. Mappings induced in homology. Topological invari= ance of singular homology30.10 - Subcomplex and quotient complexes. Factorization and isomorphism= theorems for chain complexes. Relative homology. Exact sequences. Short ex= act sequences - definition and Lemma 11.405.11 - Long exact homology sequence induced by a short exact sequence o= f chain complexes. Boundary homomoprhism of this exact sequence and its nat= urality.06.11 - Long exact homology sequence of the pair of Delta-complexes (K, = L). Long exact homology sequence of the pair of topological spaces (X,= A) and its naturality with respect to continuous mappings of pairs. Five L= emma. Splitting exact sequences. 12.11 - Chapter 12 (from Part III). Singular homology and path component= s. Zeroth homology group. Augmentation. Reduced homology groups. Homology o= f a singleton space. 13.11 - Chapter 13 - Homotopy axiom of singular homology and its consequ= ences. Excision property - formulation and calculation of homology groups o= f S^n assuming excision19.11 - Proof of excision property through Theorem 14.6. Corollary 14.5.=  20.11. - Equivalence of singular and simplicial homology theories26.11 - Examples 15.3-15.5. Homology and composition of pathes (Lemma 9.= 8. and its consequences). Mayer-Vietoris Sequences. NOTE Example 16.10 and = Proposition 16.12 are skipped for now on, we might come back to them later.= 27.11 - Brouwer's fixed point Theorem 17.1, Jordan Separation Theorem 17= .5 (and technical lemmas in between). 3.12 - Invariance of Domain. Manifolds. Degree of a mapping. Hairy Ball = Theorem4.12 - Suspension. Fundamental Theorem of Algebra. About CW complexes

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