An detail in proving the homotopy invariance of homology

[kakarotyjn]

I'm reading Allen Hatcher's topology book.In order to prove a theorem about homotopic maps induce the same homomorphism of homology groups,given a homotopy [tex]F:X \times I \to Y[/tex] from f to g,the author construct a prism operators
[tex]P:C_n (X) \to C_{n + 1} (Y)[/tex] by [tex]P(\sigma ) = \sum\nolimits_i {( - 1)^i F \circ (\sigma \times 1)|[v_0 ,...,v_i ,w_i ,...,w_n ]} [/tex] for [tex]\sigma :\Delta ^n \to X[/tex],where [tex]{F \circ (\sigma \times 1)}[/tex] is the composition [tex]\Delta ^n \times I \to X \times I \to Y[/tex].

I don't understand how sigma*1 acts on the n+1 simplex,sigma acts on n simplex,what the 1 acts on?Why[tex]F \circ (\sigma \times 1)|[\mathop v\limits^ \wedge _0 ,w_0 ,...,w_n ][/tex] equals to [tex]g \circ \sigma = g_\# (\sigma )[/tex]

Need helps,thank you!

 

[mathwonk]

well the domain space is a product, so sigma acts on the first factor, and "1" which is apparently the identity map, acts on the second factor.

i.e. forming product spaces is a functor. two spaces X,Y get changed into the space XxY,

and two maps f:X-->Z, g:Y-->W get changed into the map (fxg):XxY-->ZxW,

where (fxg)(x,y) = (f(x),g(y)).

 

[kakarotyjn]

Yes,'1' is the identity on the I, but what does [tex](\sigma \times {\rm{1}})|[{\rm{v}}_0 ,...,{\rm{v}}_i ,{\rm{w}}_i ,...,{\rm{w}}_n ][/tex] means? [v0,...,v_i,w_i,...,w_n] is a n+1 simplex,what vertex of it the '1' act on?

Thank you!

 

[mathwonk]

well from its position presumably it acts on the last one. see what works.

 

[blue_raver22]

I would like to provide some clarification on the details in proving the homotopy invariance of homology.

Firstly, in order to prove the theorem, the author constructs a prism operator P which maps simplices in the n-th chain complex of X to (n+1)-simplices in the (n+1)-th chain complex of Y. This prism operator is defined as P(\sigma) = \sum_{i}^{ }{(-1)^i F \circ (\sigma \times 1)|[v_0,...,v_i,w_i,...,w_n]} for \sigma: \Delta^n \to X, where F \circ (\sigma \times 1) is the composition \Delta^n \times I \to X \times I \to Y. Here, \sigma is a n-simplex in X and 1 is the identity map on the interval I.

The notation F \circ (\sigma \times 1)|[v_0,...,v_i,w_i,...,w_n] may seem confusing, but it simply means that we are composing the map F with the simplex \sigma and the identity map on the interval I, and then restricting it to the (n+1)-simplex [v_0,...,v_i,w_i,...,w_n]. This is done in order to define the prism operator P, which is crucial in proving the homotopy invariance of homology.

Furthermore, the author also mentions that F \circ (\sigma \times 1)|[\mathop v\limits^ \wedge _0,w_0,...,w_n] is equivalent to g \circ \sigma = g_\#(\sigma). This means that the composition of F with the simplex \sigma and the identity map on the interval I, restricted to the (n+1)-simplex [\mathop v\limits^ \wedge _0,w_0,...,w_n] is equal to the map g composed with the simplex \sigma, which is the same as the induced map g_\# on the homology groups.

I hope this helps clarify some of the details in proving the homotopy invariance of homology. It is important to understand these concepts in order to fully grasp the theorem and its implications in topology.