Questions tagged [algebraic-topology]

Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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Universal Cover of wedge sums of spaces?

I was wondering if there is some general prescription for picturing the universal cover of $X \vee Y$ for nice enough $X$ and $Y$. (Say, path-connected, locally path-connected and semilocally simply connected). In the case where one space is simply…
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Question about Surface of genus g

I understand the construction of a torus from a square by pasting opposite edges of a square and also its CW structure of that. It's easy to imagine. But how to understand the construction of a surface of genus 2g from a polygon with 4g sides and…
Kannan
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A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing one in terms of the other has determinant $+1$. Set…
S. Ha
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Is there a way to establish a correspondence between the vector bundles over a torus and some kind of homotopy groups, just as we do to spheres

By introducing the "clutching function" one can relate the complex (real) vector bundles on a sphere with homotopy groups of $GL_n(\Bbb C)$ ($GL_n^+(\Bbb R)$ for oriented ones). Can we do a similar thing to torus? The motivation is that I'm curious…
Honglu
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what is the homology groups some quotient space of torus

what is the homology group for The quotient space of $S^1 \times S^1$ obtained by identifying points in the circle $S1 \times\{x_0\} $ that differ by $\frac{2 \pi}{m}$ rotation and identifying points in the circle $\{x_0\} \times S^1 $ that differ…
kpax
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Total space of vector bundle deformation retracts onto 0-section of base space

I'm trying to prove the following: Total space of vector bundle deformation retracts onto 0-section of base space. Books seem to take this fact for granted. I checked Bott Tu and Hatcher. Online people are saying this is easy. I can't figure it…
PeterM
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Homology of knot complement

I was told in a topology class that if $Y$ is a closed $3$-manifold and $K$ is a null-homologous knot in $Y$, then $H_1(Y- \nu(K)) \cong H_1(Y) \oplus \mathbb{Z}$. I'm trying to prove this assertion, but have been unsuccessful so far. I'm trying to…
Tony
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Which manifolds are zero sets of $\mathbb R^n$ valued maps

If $M$ is a smooth manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal bundle induced by $f$. My question is: which framed submanifolds are…
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Tangent bundle of P^n and Euler exact sequence

I'm thinking about the Euler exact sequence for complex projective space, and I'm a bit confused. In the topological category, one has $$ T\mathbb{P}^n \oplus \mathbb{C} \cong L^{n+1} $$ where $L$ is the tautological line bundle and $\mathbb{C}$…
Tony
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"Naive" smash products for spectra

Suppose I work in the completeley naive homotopy category of spectra, by which I mean sequences $E = (E_n)_{n = 0, 1, \dots}$ together with maps $\sigma_{E,n}: S^1 \wedge E_n \to E_{n+1}.$ We might require the $E_n$ to be CW complexes and the…
Tom Bachmann
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The fundamental group of $U(n)/O(n)$

Since $O(n)$ is a subgroup of $U(n)$, we can consider the quotient space $L_n=U(n)/O(n)$. The quotient space is homotopic to the ($n$-th) Lagrangian Grassmannian, and it is known that its fundamental group is isomorphic to $\mathbb Z$. This fact is…
H. Shindoh
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Relationship between the $2$-plane bundles over $S^2$ and $\mathbb{Z}$

I want to follow up on this answer by asking a few more questions (posting directly on the question didn't seem to "bump" the thread). I was trying to read the referenced text (Husemoller's Fiber Bundles), but I couldn't understand that much of it…
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Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
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Cohomology groups of real projective space

My question concerns the cohomology groups $H^k(RP^n,\mathbb{Z}_2)$. We know that $H_k(RP^n,\mathbb{Z}_2) = \mathbb{Z}_2$ if $0 \leq k \leq n$ and is trivial otherwise. I looked up the solution and it says that $H^k(RP^n,\mathbb{Z}_2) =…
Balerion_the_black
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let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that $\phi_U:p^{-1}(U)\longrightarrow U\times…