Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
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Is the cap product bilinear?
This is probably a stupid question, so I apologize in advance.
On p. 239 of Hatcher's book, he defines the cap product $C_k(X;R)\times C^l(X;R)\to C_{k-l}(X;R)$ for $k\geq l$, which he claims is $R$-...
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Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
user13946
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How to compute the Betti numbers of S-D for a surface S and a divisor D?
Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper $b_i(S-D)=...
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Finding a ribbon graph for a mapping class group action
Turaev defines TQFT $(T, \tau)$ in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface $\Sigma$.
This action $\epsilon$ is ...
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Topological K-theory of Bohr compactification of real numbers
I am interested in the
K-theory of the Bohr compactification $\mathbb{R}_B$ of the real numbers.
Do we have $K_0(C(\mathbb{R}_B))$ isomorphic to $K_1(C(\mathbb{R}_B))$ ?
More generally, what do we ...
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A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
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Euler characteristic of a subset of cartesian product induced by a group action
let $X$ be a CW-complex on which a finite group $G$ acts.
define
$$F=\{ (x,gx)\;|\; x\in X ,g \in G \}$$
i want to compute the Euler characteristic of $F$. I wrote $$F=\cup_{g\in G}{F_g}\;\;,\;\; ...
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Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?
Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let $\text{Gl}_{S^...
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the boundary homomorphism $[\Sigma S^{n-1},X]\to[S^n,X]$ is identity?
Given a Puppe sequence $\cdots \to S^{n-1} \to Y \to S^n(\simeq Y/S^{n-1}) \to \Sigma S^{n-1} \to \cdots$, where $Y=S^{n-1}\cup_{2\iota} e^n$ where $\iota:S^{n-1}\to S^{n-1}$ is identity,
we have a ...
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Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
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Homomorphism between the set of n-flats in $R^m$ to some manifold
I am unaware of what the formal definition of "limit" is for a sequence of flats, but for the purpose of this question her is the definition of limit that I am using:
Consider a sequence $s_1, s_2, ...
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How to show H_BM(Conf_k(D_n))=H_BM(Conf_k(I_n))
How to show H_BM(Conf_k(D_n))=H_BM(Conf_k(I_n))?
I am trying to prove that the Borel-Moore homology of n-punctured disk D_n is the same as the Borel-Moore homology of interval [0,1] but with n-points ...
