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Mathematics Stack Exchange

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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Cell complex of torus

Suppose we have a plane model for torus. We want to decompose the plane model of torus into cell complex which satisfies 1)every face of the cell complex has exactly 6 boundary edges 2)at each vertex, there are 3 faces meet at it. How to draw such…
Idonknow
  • 15,643
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votes
2 answers

Degree of continuous maps from S1 to S1 - Two equivalent properties

I understand what is meant by the degree of a continuous map $f$ from $S^1$ to $S^1$. If we let $[S^1, S^1]$ denote the set of homotopy classes of continuous maps from $S^1$ to $S^1$, it turns out that the degree map gives a bijection from $[S^1,…
suncup224
  • 2,664
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1 answer

Quotient of a Riemannian manifold by a non-free group action

Take the example of $\mathbb{R}^2$ acted on by $C_n$ via a rotation of an angle $2 \pi/n$ around the origin. The quotient is a cone whose apex $V$ is the image of the origin. I have two questions: If we think of this as the quotient of a Riemannian…
Mr. Chip
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A question on the isomorphism induced by a homotopy equivalence.

Now, I am learning a proof that a homotopy equivalence induces an isomorphism. However, since I am a beginner in algebraic topology, I cannot fully understand the proof. Suppose $\varphi:X\to Y$ is a homotopy equivalence. To prove…
YYF
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1 answer

Simplex and face map

Let $F_{i,p}: \Delta_{p-1} \rightarrow \Delta_{p}$ denote the $i$-th face map i.e. the map that maps $e_0 \mapsto e_0$,$\dots$,$e_i \mapsto e_{i+1}$,$\dots$,$e_{p-1} \mapsto e_p$. Let's consider $\Delta_2$ (the triangle) and let its vertices be…
Rudy the Reindeer
  • 46,359
3
votes
2 answers

If all paths with the same endpoints are homotopic, then the space is simply connected.

Let $X$ be a path connected space such that any two paths in $X$ having the same end points are path homotopic. Then prove that $X$ is simply connected. I am totally stuck on this problem. Can someone help me please? Thanks for your time.
steven
  • 31
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1 answer

Fundamental group of a certain torus-like surface

Consider the wedge sum of two circles, $a V b$. Attach a 2-cell along the loop $aba^{-1}b^{-1}$.So we get a torus: From this resulting torus, remove one open disk. To this surface, identify the boundary circle where our disk used to be, with the…
Dedalus
  • 3,940
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1 answer

Retraction of surface of genus $g$

This is an exercise in 53 page of Hatcher's book : Consider surface of $M_g$ of genus $g$ If $g=h+k$ then $$M_g = M_h'\cup_{S^1} M_k'$$ where $M_h' = M_h - D^2$ Question 1 : Then show that $M_h'$ does not retract onto $S^1$ [ Hint : Abelianize…
HK Lee
  • 19,964
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1 answer

Cohomology groups of a homotopy fiber

I am reading the following: http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf and on page 316 there is a thing that gets me confused: Consider the following situation: Assume that we know that $H^\ast(K(\mathbb{Z},n),\mathbb{Q}) =…
user101036
  • 818
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0 answers

Relative Hopf degree theorem

If $f,g$ are two maps from $(D^n,S^{n-1})$ to $(D^n,S^{n-1})$ such that they have the same degree, that is $f_*[\mu]=g_*[\mu]$ where $[\mu]$ is a generator of $H_n(D^n,S^{n-1})$, then can we find a homotopy $f_t:(D^n,S^{n-1})…
Summer
  • 6,893
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2 answers

Equivalent definitions of "evenly covered"

I am just starting to learn Algebraic Topology and it would be very helpful to know whether the following two definitions are equivalent... Let $X$ and $Y$ be topological spaces and $p:Y \to X$ be a surjective continuous map. (1) An open set $U…
AlgTopBeginner
  • 31
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2 answers

Homotopy/Homology groups of rationals

Let $\mathbb{Q}$ be the set of rational numbers endowed with a topology (let's say subspace topology). What could we say about its fundamental $\pi_{1}$ and homology groups? Presumably, $\pi_{0}(\mathbb{Q}) = \mathbb{Q}$ since rationals are totally…
johnny
  • 2,263
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1 answer

Computing the fundamental group of a cell complex

In Hatcher on page 84 there is the following proposition: For a connected graph $X$ with maximal tree $T$, $\pi_1 (X)$ is a free group with basis the classes $[f_\alpha]$ corresponding to the edges $e_\alpha$ of $X - T$. I tried to apply this to the…
Rudy the Reindeer
  • 46,359
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0 answers

Proof that continuous $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ represents an injection on homology

I'm looking to prove that continuous functions $f: S^1 \rightarrow S^1$ satisfying $f(x) = -f(-x)$ for all $x \in S^1$ represent injections on homology. I'm trying to prove this fact on the way to proving the Borsuk-Ulam theorem, and I really don't…
user95612
  • 71
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2 answers

Homology and cofibrations

Please help me to answer my question.It is finding $H_*(T^2,\lbrace\ast\rbrace\times S^1\cup S^1\times\lbrace\ast\rbrace)$ using cofibrations matter.I would be grateful for your answer.
kaveh
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