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...ion with one 0-cell, $2g$ 1-cells and one 2-cell. All differentials in the chain complex are zero maps. ...ell, $h$ 1-cells and one 2-cell. The differential $C_1\to C_0$ is the zero map, while the differential $C_2\to C_1$ with respect to the basis given by the

...s given by all singular simplices, i.e.: for a topological space $X$ and a chain $c = \sum_{j=0}^{k} a_j \cdot \sigma_j \in C_*(X;\mathbb{R})$ (in reduced f If $f \colon X \longrightarrow Y$ is a continuous map of topological spaces and $\alpha \in H_*(X;\mathbb{R})$, then

...artial_s\colon C_s(T) \to C_{s - 1}(T)$ the extension over $C_s(T)$ of the map taking an $s$-dimensional cell $\sigma$ of $T$ to the boundary of $\sigma$. ...) definition as follows. Take a $k$-chain $x\in C_k(N;\Zz)$ and an $(n-k)$-chain $y\in C_{n-k}(N;\Zz)$ which are ''transverse'' to each other. The signed co

...angle = \Zz [T]/ \langle 1 + T + \cdots + T^{N-1} \rangle$. The projection map $\Zz G \rightarrow R_G$ fits into the arithmetic square: ... is true for the [[Fake lens spaces#The join construction / The suspension map|suspension]] manifold structure $\Sigma (h)$. The invariant $\mathbf{s}_{4i

...l class is represented by a degree 1 map $S^3 \rightarrow \tilde{M}$. This map induces isomorphisms on the homology and on the fundamental group. Hence it ...$M$ is always aspherical (TODO ref). The sphere theorem states, that every map $S^2\rightarrow M$ is homotopic to an embedding; and - as $M$ is irreducibl

open cube $Q\subset\Bbb R^q$ ($q = \text{codim}\, \mathcal{F}$) ''via'' the map the '''holonomy map''' $h_{VU}: D_{VU}\to T_V$,

* [[Normal map]] * [[Umkehr map]]

{{beginthm|Proposition}} Let $f:M\rightarrow X$ be a degree 1 normal map from a $2k$-dimensional (resp. $(2k+1)$-dimensional) manifold to a geometr Let $R$ be a ring with involution and $C=C_\ast$ a finite chain complex of finitely generated projective (left) $R$-modules.

and the partial evaluation map is defined as Recall, we define the cap product on the chain level by $$a\cap z := E(a\otimes \tau(z))$$ and this descends to a well def

Let $ C_* $ and $ D_* $ be $ \Zz\pi $-chain complexes and ...s y \in \Zz^{\omega} \otimes_{\Zz \pi}(C_{n-k} \otimes_{\Zz} D_k) $ to the map

...n+k} \rightarrow X_+ \wedge \text{Th} (\nu_{X})$ the canonical $S$-duality map. Choose the Thom class $u(\nu_{X}) \in C^{k}(\text{Th} (\nu_{X}))$ and the ...h (u (\nu_{X})) \sim [X]$. Prove that the following diagram commutes up to chain homotopy:


Let $F : \Aa \rightarrow \Aa'$ be a functor of additive categories with chain duality. Show that the assignment induces a $\Zz_2$-equivariant chain map

# [[Surgery obstruction map I (Ex)]] - [[User:Martin Olbermann|Martin Olbermann]] # [[Forms and chain complexes I (Ex)]]

Let $(f,b) \colon M \rightarrow X$ be a degree one normal map of $n$-GPC. is the map induced by $\rho_M$ and $\rho_X$ and denote $j \colon M \sqcup

... map of chain complexes. Define the ''algebraic mapping cone of'' $f$ as a chain complex $Cone(f)$ given in degree $k$ by

The orientation character $w\colon\pi_1(M)\to C_2$ induces a map $M \to BC_2$ from $M$ to the classifying space of $C_2$, ...ique up to homotopy. By the definition of the orientation character, this map classifies the orientation covering.

... obstruction which corresponds to the question of when a degree one normal map is normal bordant to a homotopy equivalence. Introducing surgery and the su ... many surgeries on our degree one normal map to obtain a degree one normal map covering a homotopy equivalence.

...$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$ to~$0$. Then the formula gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram

...$ be chain maps and let $j \colon gf \simeq 0 \colon C \rightarrow E$ be a chain homotopy from~$gf$ to~$0$. Then the formula gives another chain map $\Phi_j \colon \mathcal{C} (f) \rightarrow E$ making the following diagram

...uces a unique base point preserving covering map which, in turn, induces a map $f_{\ast}: C(\widehat X) \rightarrow C(\widehat Y)$ ensuring that $C(\wideh ...ms of order $(k+1)$ introduced in {{cite|Baues1991}} to realize $\xi$ by a map $\overline f: Y \rightarrow X$ with $\tau_+(\overline f) = f$.