All Questions
Tagged with homology cw-complexes
8 questions
36
votes
2
answers
4k
views
Maps which induce the same homomorphism on homotopy and homology groups are homotopic
I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...
- 1,129
9
votes
1
answer
905
views
A description of cellular boundary maps in terms of a Morse function
I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...
- 319
5
votes
1
answer
654
views
loop space of a finite CW-complex
Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups ...
- 855
5
votes
1
answer
384
views
A weak version of the Whitehead Theorems
Let $f:X\longrightarrow Y$ be a map between CW-complexes $X$ and $Y$. By the Whitehead Theorems, if one of the conditions:
1- (homotopy version) $\pi_n (f):\pi_n (X)\longrightarrow \pi_n (Y)$ is an ...
- 1,182
4
votes
1
answer
479
views
Cellular homology of the universal cover
Let $X$ be a connected pointed CW complex. Let $\tilde{X}$ be its universal covering space and $G=\pi_{1}(X)$.
Lets denote $(C^{Cell}_{\ast}(\tilde{X}),d)$ the cellular chain complex associated to $\...
- 855
2
votes
0
answers
294
views
What topological spaces can be realized as cell complexes?
What are the topological spaces can be realized as cell complexes, up to homeomorphism? It seems for instance that all manifolds can be built from cell complexes. It is clear however that one can ...
- 504
2
votes
0
answers
345
views
regular CW complex and incidence matrices
Suppose that we have a regular CW-complex $X$. I want to define the incidence matrix of $k$-skeleton of $X$ with respect to the $k-1$ skeleton and I wonder what might go wrong in this case.
If it's ...
- 504
1
vote
0
answers
469
views
Cellular chain complex of $G$-CW-complexes & their differentials
I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of
$G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf
...
- 5,998