# Algebraic topology from geometric viewpoint

@article{Skopenkov2008AlgebraicTF, title={Algebraic topology from geometric viewpoint}, author={Arkadiy Skopenkov}, journal={arXiv: Geometric Topology}, year={2008} }

This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and invariants, characteristic classes). Thus main ideas of algebraic topology are presented with minimal technicalities. Familiarity of a reader with basic notions of topology (such as 2-dimensional manifolds and vector fields) is desirable, although definitions are given…

## 13 Citations

A user's guide to the topological Tverberg conjecture

- Mathematics
- 2018

A simplified explanation of easier parts of the arguments of the Tverberg conjecture is presented, accessible to non-specialists in the area, and reference to more complicated parts are given.

G T ] 2 5 M ay 2 01 8 Invariants of graph drawings in the plane

- Mathematics
- 2018

We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional…

A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds

- MathematicsArXiv
- 2021

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short…

Ju l 2 02 1 A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2 k-manifolds ∗

- Mathematics
- 2021

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short…

On the Kuratowski graph planarity criterion

- MathematicsArXiv
- 2008

This paper presents the Makarychev proof (with further simplifications by Prasolov, Telishev, Zaslavski and the author) which is possibly the simplest proof of the Kuratowski graph planarity criterion.

Criteria for integer and modulo 2 embeddability of graphs to surfaces

- Mathematics
- 2020

The study of graph drawings on 2-surfaces is an active area of mathematical research and the main results are criteria for Z2- embeddability and Z-embeddability of graphs to surfaces (Theorems 1.1 and 1.4).

A user's guide to basic knot and link theory

- MathematicsTopology, Geometry, and Dynamics
- 2021

We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible to a…

Extendability of simplicial maps is undecidable

- MathematicsArXiv
- 2020

A gap is exhibited in the Filakovský-Wagner-Zhechev proof that embeddability of complexes is undecidable in codimension $>1$ and there is no algorithm recognizing the extendability of the identity map of S^l\vee S* to a PL map.

Towards a short proof of the Fulek-Kynčl criterion for modulo 2 embeddability of graphs to surfaces

- MathematicsArXiv
- 2020

A general position PL map g : K → M (a.k.a. a graph drawing on M) is called a Z2(almost) embedding if |gσ ∩ gτ | is even for any pair σ, τ of non-adjacent edges so that the graph K to Sg is homeomorphic to the sphere with g handles and a hole.

Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

- MathematicsArXiv
- 2021

This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.