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Results tagged with homology
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user 318
Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
2
votes
Euler class of vertical tangent bundle of the surface bundle over circle
Let me not Poincare dualise, and work in cohomology.
Let me generalise the setting you have described, and consider the universal surface bundle $\pi : E \to M_g^1$ over the moduli space of surfaces w …
- 18.2k
11
votes
Accepted
What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?
Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Le …
- 18.2k
7
votes
Accepted
The Image of the Mod 2 Homology of BSp in the Homology of BSO
I don't know how the homology of $BSO$ is best described inside that of $BO$, but presumably one can find out. …
- 18.2k
14
votes
A map inducing isomorphisms on homology but not on homotopy
It is a 3-manifold which has the homology of $S^3$, but non-trivial fundamental group (the binary icosahedral group). … In particular, if we remove a point from $M$ we obtain a space $X$ which has the homology of a point (one can verify this by Mayer--Vietoris) but non-trivial fundamental group. …
- 18.2k
2
votes
Accepted
Homology dimension of the mapping class group of a surface with boundary
As the homology of $\mathcal{M}_g$ vanishes in degrees at least $6g-7$, and $\mathbb{S}(\Sigma_g)$ is a 3-manifold, the Serre spectral sequence implies that the homology of $\mathcal{M}_{g}^1$ vanishes … Iterating, the homology of $\mathcal{M}_{g, n}^b$ vanishes in degrees at least $6g-7+2n+3b$. …
- 18.2k