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Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old …

Consider a compact Riemann surface $X$ of genus $g$. It is well-known that its fundamental group $\pi_1(X)$ is the free group on the generators $a_1,b_1,...,a_g,b_g$ divided out by the normal subgrou …

Let $M$ be a smooth manifold and $E$ a smooth real vector bundle of even rank over $M$. If $E$ admits of a complex vector bundle structure $\mathcal E$ ($\mathcal E_\mathbb R=E$) then all odd Stiefel- …

Dear Eric, here is a Bourbaki-style proof. Recall that a continuous map $f: Y\to X$ is called proper by Bourbaki if, for all spaces $Z$, the map $f\times 1_Z: Y \times Z\to X \times Z$ is closed. …

Griffiths and Morgan wrote a fine book on the subject. Apart from the obvious attractiveness of learning a theory from its creator, it is written in an amazingly user-friendly style. For example, Chap …

Blow-up $\mathbb C^2$ simultaneously at all points of $\mathbb Z \times \{0\}$. Part a) is evident for the blown-up manifold $X$ since it contains one-dimensional compact submanifolds. As for …

Dear Jankir, 1) The book by Sergey V. Matveev Lectures on Algebraic Topology, Sergey V. Matveev (EMS Series of Lectures in Mathematics, 2006) contains about 10 pages of hints and solutions to i …

Yes, if $G$ is an abelian group, its classifying space $BG$ is an abelian topological group whose $\pi _1$ is $G$. You can find details in John Baez's wonderful post, hearteningly called "Classifyi …

Consider the standard embedding of the unit interval in $\mathbb R^2$ viz. $I=[0,1]\times \{0\} \subset \mathbb R^2$. Let $C$ denote the Cantor subset $C \subset I$ and define $U= \mathbb R^2 - C$, an …

Serre and Cartan wrote a Comptes Rendus note Un Théorème de finitude concernant les variétés analytiques compactes where they proved that all cohomology complex vector spaces $H^q(X,\mathcal F)$ of an …

Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$. Question: Is the top singular cohomology group $H^n(M,\mathbb Z)$ zero? This naïve question does not seem to …

Ad question 1): Yes, all open star-shaped subsets of $\mathbb{R}^n$ are diffeomorphic to $\mathbb{R}^n$. This is surprisingly little-known and there is a proof due to Stefan Born. You can find t …

"...utinam intelligere possim rationacinationes pulcherrimas quae e propositione concisa DE QUADRATUM NIHILO EXAEQUARI fluunt." Henri CARTAN [...if I could only understand the beautiful consequences …

An interesting application of De Rham's theorem is to show that certain differential manifolds are not diffeomorphic. Here are two examples. 1) For $n$ even the sphere $S^n$ and real projective spac …

Dear Timothy, here is a theorem which, according to your wish, "could be understood, and seen to be interesting, by someone who had not already studied the material in that course": Brouwer's celebrat …