An detail in proving the homotopy invariance of homology

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In summary, the author of Allen Hatcher's topology book discusses the construction of prism operators in order to prove a theorem about homotopic maps inducing the same homomorphism of homology groups. These operators are defined as P(sigma) = sum of (-1)^i times the composition of F and (sigma x 1) restricted to a specific n+1 simplex. The identity map acts on the second factor of the product space, while sigma acts on the first factor. This is because forming product spaces is a functor.
  • #1
kakarotyjn
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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!
 
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  • #2
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)).
 
  • #3
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!
 
  • #4
well from its position presumably it acts on the last one. see what works.
 
  • #5


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.
 

Related to An detail in proving the homotopy invariance of homology

1. What is homotopy invariance of homology?

Homotopy invariance of homology is a property of topological spaces that states that homology groups are invariant under continuous deformations of the space. This means that if two spaces are homotopy equivalent, meaning they can be continuously transformed into each other, then their homology groups will be isomorphic.

2. Why is homotopy invariance important in topology?

Homotopy invariance is important because it allows us to classify and distinguish between topological spaces. By studying the homology groups of a space, we can determine its topological properties and understand how it is related to other spaces. It also allows us to prove theorems and make mathematical statements about topological spaces.

3. How is homotopy invariance of homology proven?

The proof of homotopy invariance of homology involves using the axioms and properties of homology groups, as well as techniques from algebraic topology such as the Mayer-Vietoris sequence and the long exact sequence of a pair. The proof also relies on the concept of chain homotopies, which are continuous transformations between chain maps.

4. What is the significance of proving homotopy invariance of homology?

Proving homotopy invariance of homology is significant because it is a fundamental result in algebraic topology that has many applications. It allows us to study and classify topological spaces, as well as establish connections between different areas of mathematics such as algebra, geometry, and topology.

5. Can homotopy invariance be extended to other algebraic structures?

Yes, homotopy invariance can be extended to other algebraic structures such as homotopy groups and cohomology groups. These structures also have the property of being invariant under homotopy equivalences and play an important role in understanding the topology of spaces.

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