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comment A Künneth-Theorem version for relative singular cohomology
It is true, and is proven in the same way with notational changes: Corollary 12.10 of Dold's Lectures on Algebraic Topology.
Jan
29
comment Is there any work on “super Fukaya categories”?
An immediate subquestion is whether there is Morse theory for supermanifolds.
Jan
19
awarded  Yearling
Jan
8
comment Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57)
Also here: mathoverflow.net/questions/5518/…
Jan
8
comment Homotopy classes of maps and cohomology classes (Hatcher, AT, Thm 4.57)
This was answered here: mathoverflow.net/questions/2890/… To repeat: $H^n(K(G,n);G)=Hom(\pi_nK(G,n),G)=Hom(G,G)$ and so there is a distinguished element $u\in H^n(K(G,n);G)$ corresponding to the identity $1:G\to G$. The bijection is given by pull-back, $f\mapsto f^*u$. For your simple example ($G =\mathbb{Z}$, $n = 1$), take $c \in H^1(X)$ to map the 1-skeleton of $X$ to $S^1$, where an edge $e$ will make $c(e)$ loops around $S^1$.
Dec
15
comment Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?
In case anyone is interested, Theorem F of that paper also gives the analog for $M^2\to\mathbb{R}^3\to\mathbb{R}^4$. There is a nice example (due to Giller) for which Koschorke's theorem doesn't help: The oriented double cover of Boy's surface is an immersion of $S^2$ into $\mathbb{R}^3$ that cannot be "lifted" to an embedding in $\mathbb{R}^4$.
Dec
14
comment Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?
Is this true for $M^2\to\mathbb{R}^3\to\mathbb{R}^5$?
Nov
21
comment In what sense is the braid group $B_3$ the universal central extension of the modular group $\Gamma$?
It has something to do with the fact that it sits (as a lattice) inside the universal cover of $PSL_2(\mathbb{R})$ (as topological group), and we have the commutative diagram of short exact sequences of groups (with the same factor of $\mathbb{Z}$). Googling more, there is this book Moonshine beyond the Monster which argues that $B_3$ is "universal" in some sense, and Baez tries to explain it in his blog post here: math.ucr.edu/home/baez/week233.html
Nov
3
revised Interpretation of universal coefficients theorem for group cohomology
rephrased what I wrote to make my "point" clearer
Oct
31
awarded  Popular Question
Oct
27
revised irreducibility of discriminant
Incorrect subscript
Oct
23
comment How to compute second homology of a group given by presentation with two relators
And in that same chapter of the book, you'll find more results for one relator groups, if that interests you.
Oct
23
answered How to compute second homology of a group given by presentation with two relators
Oct
16
awarded  Notable Question
Sep
23
revised Non-realizability of $\mathbb{Q}$ as a cohomology group
added 249 characters in body
Sep
23
answered Non-realizability of $\mathbb{Q}$ as a cohomology group
Sep
18
awarded  Popular Question
Sep
17
comment electron configuration on manifolds
I asked this question a long time ago, for both discrete and continuous charge distributions: mathoverflow.net/q/80731/12310 (I originally voted to close as a result, but our questions do seem to differ in the sense that I care about large k where you care about any k).
Sep
12
comment Question about theorem in Arnold's book on action-angles variables
The Inverse Function Theorem will give coordinates $(0,q)$ on $M_f\subset \mathbb{R}^{2n}$. We also know that $(I,\phi)$ are the coordinates for the neighborhood of $M_f$ (in $\mathbb{R}^{2n}$). That's what Arnold is using.
Sep
6
comment Hodge de Rham operator and orientability
George de Rham does this in his book Differentiable Manifolds. He develops the theory without restricting to orientable manifolds: he uses the language of "even" and "odd" forms. The Hodge-star of an even $k$-form is odd (and vice versa).