Lorenzo Mantovani

Algebra and Topology

(Mat 575, Spring Semester 2020)

Lecturers: C. Dahlhausen and L. Mantovani
Tutor: E. Jacobsen

Introduction

The main object of study of this course will be that of a sheaf on a topological space $X$. Actually a more informative subtitle of the course could be: An introduction to algebraic topology via sheaf theory. The notion of sheaf, to my knowledge, is attributed to J. Leray and dates back to the Forties. In the Fifties several mathematicians, building upon the work of Leray, developed sheaf theory after realizing it could be applied to a huge variety of contexts. In some cases shaves are actually the common ground where apparently unrelated topics share some common features. Such a wide range of applications was made possible by the work of H. Cartan, J.-P. Serre and A. Grothendieck, just to name the fist coming to mind. See the following link for an historical introduction to the theory (in French).

Giving the proper definition of sheaf here would take a bit too long and would probably not be very informative at this stage. Informally, however, we can think of a sheaf $\mathcal F$ on a topological space $X$, as a collection of local data on $X$ that can be glued from an open cover of $X$. For instance: the collection of continuous real-valued functions on $X$ is a sheaf. Indeed a function $f: X \rightarrow \mathbb R$ is continuous if and only if it continuous in a neighbourhood of each point $x$ of $X$. A non example is the collection of $\mathbb R$-valued integrable functions on $X:=\mathbb R$. Indeed a non-zero constant function is integrable on some open cover of $X$ but not on the whole $X$.

The first part of the course will thus be devoted to an introduction, from scratch, to the notion and the properties of sheaves on a topological space $X$. This will include also some basic algebra and category theory, which are both needed to handle sheaves properly and with satisfaction.

The second part of the course will be devoted to one possible application of sheaf theory: constructing and studying the main properties of singular cohomology. Singular cohomology has been studied since the Thirties. Once again this is not the right place for a formal definition. Given a sufficiently nice topological space $X$, the Singular Cohomology groups of $X$ with coefficients in $\mathbb Z$ are a collection of abelian groups $H^i(X,\mathbb Z)$ that give one possible answer to the following vague problem.

Associating with a (nice) topological space $X$ an algebraic invariant $H(X)$,
in such a way that, at the same time, we can:
manipulate the invariant $H(X)$ in a sufficiently easy and intuitive way,
and reconstruct some properties of $X$ from $H(X)$.

As vague as it might seem, this problem has been motivating algebraic topology for several decades (and still does), and in addition singular cohomology is the best understood algebraic invariant of topological spaces. It is of capital relevance in the study of manifolds, differential forms, Hodge theory, and even differential equations.

During the last part of the course we will actually touch with our hands some of the applications of the theory. For instance, on the manifolds point of view, we will see the relation between singular cohomology and differential forms. From the point of view of algebraic topology, we will review the theory of covering spaces and outline its relation with $\pi_1$-representations and with locally constant sheaves of finite sets. If the time permits, depending on taste of the audience, we might see some more applications of the theory to other areas of mathematics such as connections and ordinary differential equations or to Galois theory.

Program in short

Board Diary

18.02.2020 - Introduction to the course: motivation, aims and applications. Introduction to categories and functors. See 1.3 of [AT]

20.02.2020 - Natural transformations and equivalences of categories. Fully faithfulness + essential surjectivity $\Leftrightarrow$ Equivalence. Presheaves of sets on a category and the Yoneda Lemma.

25.02.2020 - Yoneda embedding, representable functors. Initial and Final objects, unions and products.

27.02.2020 - Limits and colimits and some of their properties: (co)limits in two variables, (co)limits with values in functor categories. Presheaves admit (co)limits indexed on small categories.

03.03.2020 - Adjunctions. Functors commuting with limits.

05.03.2020 - Filtered colimits. Filtered colimits in sets commute with finite limits. "Re-indexing" filtered colimits using cofinal functors. The forgetful functor from $A$-Modules to sets preserves filtered colimits. A fundamental example: the "stalk" (see next lectures) $\mathcal C^\infty_{X,x}$ of the (pre-)sheaf of real-valued infinitely-differentiable functions on $X=]-1,1[$ in $x=0$. $\mathbb R$-linear $\mathbb R$-valued derivations on $\mathcal C^\infty_{X,x}$ form a one-dimensional vector space over $\mathbb R$ generated by $\frac{\operatorname d}{\operatorname d t}$ (the tangent space in $0$).

WARNING: From this point on, due the recent concerns about the spread of viral infections the lectures will take place in a on-line form. Contact me in order to know how to participate.

10.03.2020 - Revision of pre-sheaves of sets and of $A$-modules on a topological space $X$. Examples. How to play with pre-sheaves: taking sections over an open subset, restricting to the filter of open neighbourhoods $I^{op}_x$ of a point $x \in X$. Push-forward along a map. These three operations are functorial and exact, i.e. they commute with finite limits and colimits, and actually with all limits and colimits. Sheaves and maps of sheaves. Some examples.

12.03.2020 - No class.

17.03.2020 - Alternative equivalent definitions of sheaf. Stalks of a (pre)-sheaf. Behaviour of monos end epis under the operation of taking stalks in the categories of pre-sheaves and of sheaves of sets and $A$-modules. Sheafification: the forgetful functor $\operatorname{Shv}(X) \subseteq \operatorname{Psh}(X)$ has a left adjoint $a(-)$ called sheafification. Construction of $a(-)$ and properties, as in Theorem 5.3.3 in [AT].

19.03.2020 - Four operations on sheaves of $A$-modules on $X$. Tensor product $-\otimes_A-$ of sheaves and hom sheaf $\mathcal{Hom}_X(-,-)$ and the adjunction $\big (-\otimes_A-,\mathcal{Hom}_X(-,-) \big)$. Pull-back $f^{-1}$ and push-forward $f_\ast$ associated with a morphism of sites (e.g. a continuous map) $f: X \rightarrow Y$, and the adjunction $(f^{-1},f_\ast)$ associated with $f$.

24.03.2020 - Examples of pull-back and push-forward associated to some special maps: projection to the point, inclusion of a point, inclusion of open and closed subsets. Notions of homological algebra in sheaves of $A$-modules on a space $X$: $\operatorname{Ker}$, $\operatorname{Coker}$, $\operatorname{Im}$, $\operatorname{Coim}$. $\operatorname{Im}= \operatorname{Coim}$ in $\operatorname{Sh}(X,A)$, exact complexes. Sections of a sheaf on an arbitrary subset $S\subseteq X$, and properties of the adjunction $(i^{-1},i_\ast)$ induced by the inclusion $i: S \subseteq X$.

26.03.2020 - Some remarks on monos and epis of sheaves of $A$-modules on a space: description in terms of points and sections. Monos and epis are local notions. The adjunction $(i^{-1},i_\ast)$ induced form the inclusion of a closed subset $i:Z \subseteq X$ and its main properties.

31.03.2020 - The adjunction $(i_!,i^{-1},i_\ast)$ induced form the inclusion of an open subset $i:U \subseteq X$ and its main properties. Gluing sheaves, gluing philosophy: local properties of sheaves glue. Examples: sheaves on spheres, an overview on sheaves on manifolds.

02.04.2020

Exercises and Homeworks

There are both exercises and homeworks. Every week there will appear either an exercise sheet or a homework sheet.

Exercises are an invitation to think about the lectures material and they will be discussed in the exercise class. They are voluntary and you do not have to hand in written solutions. However, we encourage you to write up solutions for yourself and to present some of them once in a while in the exercise class. The list of exercises will be consistently updated, and will be available on the official page of the course.

Homeworks are mandatory problems we expect you to work on. You should hand in written solutions in a readable and well-structured form. There will be five homework sheets during the semester. For each of them you have two weeks time to submit your solutions. They will be graded so that you get feedback. In order to take part in the final exam you have to obtain at least 50% of the maximal points of all homework sheets. Homeworks will also be consistently updated and be made available on the official page.

Final Exam

There will be a final exam. Depending on the number of participants, it will be either in a written form or an oral examination. In order to be allowed to take part in the exam you have to obtain at least 50% of the maximal points of all homework sheets.

References

The main references for this course are the following books and lecture notes. What we cover during the course is a strict subset of the content of such books, but we often go into some more details and computations.

For some more applications of the theory to Analysis and to Covering/Galois Theory one can give a look respectively at

Here are instead some other references on the subject:

Links

The Link to the corresponding module on the course catalog of the University.