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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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21356 questions
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Long exact sequence of a fibration, center

Let $p:E \rightarrow B$ be a fibration with fiber $F$ . Associated to this we have a long exact sequence $$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \cdots.$$ I am trying to show that…
user161954
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Isomorphic fundamental groups result in homeomorphism?

I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic?
square
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Question in Hatcher

Exercise 0.21 of Hatcher's Algebraic Topology reads: If $X$ is a connected Hausdorff space that is the union of a finite number of $2$-spheres, any two of which intersect in at most one point, show that $X$ is homotopy equivalent to a wedge sum of…
Jonathan Gleason
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Are homotopic maps over a cofibration homotopic relative to the cofibration?

Let $X$ be a Hausdorff space and $A$ a closed subspace. Suppose the inclusion $A \hookrightarrow X$ is a cofibration. Let $f, g: X \to Y$ be maps that agree on $A$ and which are homotopic. Are they homotopic relative to $A$? My motivation for asking…
Akhil Mathew
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If a manifold M has zero Euler characteristic, there is a non-vanishing vector field on it

There is hint: if M has isolated singular points, find a diffeomorphism to make these singular points in a any neighborhood which you want. How can we do next?
henry
  • 309
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1 answer

Why is there no compact manifold without boundary with the following homology groups?

I've been studying homology groups, and this question is stumping me: Prove there can be no compact manifold $X$ without boundary whose homology groups are $$H_i(X) = \left\{ \begin{array}{ll} \mathbb{Z} & i = 0 \\ \mathbb{Z}_3 & i=1 \\ 0 & i = 2 \\…
phenomenalwoman4
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Weak deformation retraction (exercise 0.4 from Hatcher) - Proof check

A deformation retraction in the weak sense of a space $X$ to a subspace $A$ is a homotopy $f_t:X\to X$ such that $f_0=Id_X$, $f_1(X)\subset A$ and $f_t(A)\subset A$ for all $t$. Show that if $X$ deformation retracts to $A$ in this weak sense, then…
Xena
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Wedge product $S^1 \vee S^2$

I am trying to compute $\pi_1(S^1 \vee S^2$) by Van Kampen. I know Hatcher has a solution but I need to verify if my approach is correct and rigorous. I have seen a previous post on this topic, but I am using a different decomposition of $X=S^1 \vee…
nonlinearism
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deformation retract and strong deformation retract

I am trying to gain some intuition about retracts, deformation retracts and strong deformation retracts (see http://en.wikipedia.org/wiki/Deformation_retract for definitions). We have that any strong deformation retract is a deformation retract and…
MBL
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Identifying a $\Delta$ complex

I am doing some self study and am having trouble with the following. I want to say the answer is a cone, but I do not think that this is correct. Help will be apreciated. What familiar space is the quotient $\Delta$ complex of a 2 simplex $[v_0 ,…
dinky
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Can a finite group act freely (as homeomorphisms) on $\mathbb R^n$

I am asking if whether or not a finite group acts freely (as homeomorphisms) on $\mathbb R^n$. To answer in the negative, it suffices to show: for any homeomorphism $f$ such that $f^d=\text{id}_{\mathbb R^n}$, then $f$ has a fixed point. I am…
Ash GX
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Proof of another Hatcher exercise: homotopy equivalence induces bijection

I'm doing stuck with the first half of exercise 12 on page 19 in Hatcher: Exercise: Show that a homotopy equivalence $f : X \rightarrow Y$ induces a bijection between the set of path-components of $X$ and the set of path-components of $Y$. and that…
Rudy the Reindeer
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Proof that $X$ contractible $\iff \forall f:X \rightarrow Y$ : $f \cong const.$

I'm reading Hatcher and I did exercise 10 on page 19. Can you tell me if my answer is correct? Many thanks for your help! Claim: $X$ contractible $\Leftrightarrow \forall$ arbitrary maps $f:X \rightarrow Y$, $Y$ arbitrary, $f \cong…
Rudy the Reindeer
  • 46,359
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1 answer

Transition functions for the tautological bundle

Define the tautological bundle over $CP^1$ to be $\tau = \{[a_1, a_2], (z_1, z_2) \in CP^1\times\mathbb{C}^2 | \exists \lambda \in \mathbb{C} \;\text{such that} \;\lambda (z_1,z_2) = (a_1, a_2) \}.$ Then $\tau$ trivializes over open sets $U_i =…
mck
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Deformation retracting the torus minus a point to of figure 8

I am attempting to find an explicit deformation retract from the torus minus some point to a figure of eight. Thus far, I have realised it is sufficient to show that: If $I=[-1,1]$, then $I^2 - \{0,0\}$ deformation retracts to $\partial I^2$, since…
J.Jones5552
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